7. Adjoints

“Adjoint functors arise everywhere.”

——Mac Lane

We assume that our categories are locally small.

7.1Preliminary Definition

定义 7.1.1. An adjunction or adjoint pair between categories $\mathcal{C}$ and $\mathcal{D}$ consists of two functors $F:\mathcal{C}\to \mathcal{D}$ and $U:\mathcal{D}\to \mathcal{C}$ and a natural transformation $\eta :1_{\mathcal{C}}\to U\circ F$ with the following universal mapping property: for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ and for every arrow $f:C\to U(D)$, there is a unique arrow $g:F(C)\to D$ in $\mathcal{D}$ such that the following diagram is commutative:

hence $U(g)\circ {\eta }_C=f$.

Here $F$ is called a left adjoint of $U$, $U$ is called a right adjoint of $F$ and $\eta$ is called the unit of adjunction. Also, $(F,U)$ is called an adjunction. Sometimes, one denotes $F⊣U$.

注 7.1.2. For every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$, we may consider $$\Phi :{\mathrm{Hom}}_{\mathcal{D}}(F(C),D)\to {\mathrm{Hom}}_{\mathcal{C}}(C,U(D))$$given by $\Phi (g)=U(g)\circ {\eta }_C$ for every $g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D)$. By universal mapping property of adjunction, $\Phi$ is bijective.

例 7.1.3. Let $\mathcal{C}$ be a category and let $\Delta :\mathcal{C}\to \mathcal{C}\times \mathcal{C}$ be the diagonal functor defined on objects $C$ of $\mathcal{C}$ by $\Delta (C)=(C,C)$ and on arrows $f:C\to C^{\prime }$ in $\mathcal{C}$ by $\Delta (f)=(f,f):\Delta (C)=(C,C)\to \Delta \left(C^{\prime }\right)=\left(C^{\prime },C^{\prime }\right)$. We are looking for a right adjoint of $\Delta$, say $R:\mathcal{C}\times \mathcal{C}\to \mathcal{C}$. Then we must have an isomorphism in $\mathbf{Set}$: $${\mathrm{Hom}}_{\mathcal{C}}(C,R(X,Y))\cong {\mathrm{Hom}}_{\mathcal{C}\times \mathcal{C}}(\Delta (C),(X,Y))$$for every object $C$ of $\mathcal{C}$ and for every object $(X,Y)\in \mathcal{C}\times \mathcal{C}$.

We define the functor $R:\mathcal{C}\times \mathcal{C}\to \mathcal{C}$ on objects $(X,Y)$ of $\mathcal{C}\times \mathcal{C}$ by $R(X,Y)=X\times Y$ and on arrows $(f,g)$ in $\mathcal{C}\times \mathcal{C}$ by $R(f,g)=f\times g$. One shows that $(\Delta ,R)$ is an adjunction with unit $\eta :1_{\mathcal{C}}\to R\circ \Delta$, whose component at an object $C$ of $\mathcal{C}$ is $${\eta }_C=\left[\begin{array}{c}1\\ 1\end{array}\right]:C\to (R\circ \Delta )(C)=C\times C.$$For every arrow $h:C\to D$ in $\mathcal{C}$, the following diagram is commutative:

hence $\eta$ is a natural transformation.

Next one checks universal mapping property for $\eta$, namely for every objects $C$ of $\mathcal{C}$ and $(X,Y)$ of $\mathcal{C}\times \mathcal{C}$ and for every arrow $f:C\to R(X,Y)=X\times Y$ in $\mathcal{C}$, there is a unique arrow $g:\Delta (C)=(C,C)\to (X,Y)$ in $\mathcal{C}\times \mathcal{C}$ such that the following diagram is commutative:

hence $R(g)\circ {\eta }_C=f$. Since $f:C\to X\times Y$, it is of the form $f=\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]$, where $f_1:C\to X$ and $f_2:C\to Y$. Then we may choose$$g=\left(f_1,f_2\right):(C,C)\to (X,Y)$$One checks that this choice yields the required properties.

7.2Hom-set Definition

定理 7.2.1. Let $F:\mathcal{C}\to \mathcal{D}$ and $U:\mathcal{D}\to \mathcal{C}$ be functors. Then $(F,U)$ is called an adjunction if one of the following equivalent conditions holds:

(1)	(preliminary definition) There exists a natural transformation $\eta :1_{\mathcal{C}}\to U\circ F$ (called the unit of adjunction) with the following universal mapping property: for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ and for every arrow $f:C\to U(D)$, there is a unique arrow $g:F(C)\to D$ in $\mathcal{D}$ such that the following diagram is commutative: hence $U(g)\circ {\eta }_C=f$.
(2)	(official definition) For every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ there is an isomorphism in $\mathbf{Set}$: $${\Phi }_{C,D}:{\mathrm{Hom}}_{\mathcal{D}}(F(C),D)\to {\mathrm{Hom}}_{\mathcal{C}}(C,U(D))$$which is natural in both $C$ and $D$.
(3)	(dual to preliminary definition) There exists a natural transformation $\epsilon :F\circ U\to 1_{\mathcal{D}}$ (called the counit of adjunction) with the following universal mapping property: for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ and for every arrow $g:F(C)\to D$, there is a unique arrow $f:C\to U(D)$ in $\mathcal{C}$ such that the following diagram is commutative: hence ${\epsilon }_D\circ F(f)=g$.

If the inverse of ${\Phi }_{C,D}:{\mathrm{Hom}}_{\mathcal{D}}(F(C),D)\to {\mathrm{Hom}}_{\mathcal{C}}(C,U(D))$ is$${\Psi }_{C,D}:{\mathrm{Hom}}_{\mathcal{C}}(C,U(D))\to {\mathrm{Hom}}_{\mathcal{D}}(F(C),D),$$we have the following relating formulas:

•	${\Phi }_{C,D}(g)=U(g)\circ {\eta }_C$ and ${\eta }_C={\Phi }_{C,F(C)}\left(1_{F(C)}\right)$.
•	${\Psi }_{C,D}(f)={\epsilon }_D\circ F(f)$ and ${\epsilon }_D={\Psi }_{U(D),D}\left(1_{U(D)}\right)$.

证明. (1)$⟹$(2) Define ${\Phi }_{C,D}(g)=U(g)\circ {\eta }_C$. By universal mapping property from (1) we know that ${\Phi }_{C,D}$ is an isomorphism in $\mathbf{Set}$ (bijective).

We show naturality in $C$. Let $h:C^{\prime }\to C$ be an arrow in $\mathcal{C}$. We need to prove that the following diagram is commutative:

Denote $h^{\ast }={\mathrm{Hom}}_{\mathcal{C}}(h,U(D))$ and $F(h{)}^{\ast }={\mathrm{Hom}}_{\mathcal{D}}(F(h),D)$. Let $g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D)$. We have:$$\left(h^{\ast }\circ {\Phi }_{C,D}\right)(g)=h^{\ast }\left(U(g)\circ {\eta }_C\right)=U(g)\circ {\eta }_C\circ h.$$Since $\eta$ is a natural transformation, the following diagram is commutative:

It follows that:$$\begin{array}{rl}\left(h^{\ast }\circ {\Phi }_{C,D}\right)(g)& =U(g)\circ (U\circ F)(h)\circ {\eta }_{C^{\prime }}=U(g\circ F(h))\circ {\eta }_{C^{\prime }}\\ & ={\Phi }_{C^{\prime },D}(g\circ F(h))={\Phi }_{C^{\prime },D}\left(F(h{)}^{\ast }(g)\right)=\left({\Phi }_{C^{\prime },D}\circ F(h{)}^{\ast }\right)(g).\end{array}$$Hence we have naturality in $C$.

We show naturality in $D$. Let $k:D\to D^{\prime }$ be an arrow in $\mathcal{D}$. We need to prove that the following diagram is commutative:

Denote $k_{\ast }={\mathrm{Hom}}_{\mathcal{D}}(F(C),k)$ and $U(k{)}_{\ast }={\mathrm{Hom}}_{\mathcal{C}}(C,U(k))$. Let $g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D)$. We have:$$\begin{array}{rl}\left(U(k{)}_{\ast }\circ {\Phi }_{C,D}\right)(g)& =U(k{)}_{\ast }\left(U(g)\circ {\eta }_C\right)=U(k)\circ U(g)\circ {\eta }_C=U(k\circ g)\circ {\eta }_C\\ & ={\Phi }_{C,D^{\prime }}(k\circ g)={\Phi }_{C,D^{\prime }}\left(k_{\ast }(g)\right)=\left({\Phi }_{C,D^{\prime }}\circ k_{\ast }\right)(g).\end{array}$$Hence we have naturality in $D$.

(2)$⟹$(1) We have seen that naturality in $D$ means that for every arrow $k:D\to D^{\prime }$ in $\mathcal{D}$ we have:$$\left({\Phi }_{C,D^{\prime }}\circ k_{\ast }\right)(g)=\left(U(k{)}_{\ast }\circ {\Phi }_{C,D}\right)(g),\forall g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D),$$that is,$${\Phi }_{C,D^{\prime }}(k\circ g)=U(k)\circ {\Phi }_{C,D}(g),\forall g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D).$$(7.1)Also, naturality in $C$ means that for every arrow $h:C^{\prime }\to C$ in $\mathcal{C}$ we have:$$\left(h^{\ast }\circ {\Phi }_{C,D}\right)(g)=\left({\Phi }_{C^{\prime },D}\circ F(h{)}^{\ast }\right)(g),\forall g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D),$$that is,$${\Phi }_{C,D}(g)\circ h={\Phi }_{C^{\prime },D}(g\circ F(h)),\forall g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D).$$(7.2)

We show that $\eta :1_{\mathcal{C}}\to U\circ F$ with the following components at an object $C$ of $\mathcal{C}$ :$${\eta }_C:C\to (U\circ F)(C),{\eta }_C={\Phi }_{C,F(C)}\left(1_{F(C)}\right)$$is a natural transformation. Let $h:C^{\prime }\to C$ be an arrow in $\mathcal{C}$. Consider the following diagram:

We have:$$\begin{array}{rl}(U\circ F)(h)\circ {\eta }_{C^{\prime }}& =(U\circ F)(h)\circ {\Phi }_{C^{\prime },F\left(C^{\prime }\right)}\left(1_{F\left(C^{\prime }\right)}\right)=U(F(h))\circ {\Phi }_{C^{\prime },F\left(C^{\prime }\right)}\left(1_{F\left(C^{\prime }\right)}\right)\\ & \stackrel{(7.1)}{=}{\Phi }_{C^{\prime },F(C)}\left(F(h)\circ 1_{F\left(C^{\prime }\right)}\right)={\Phi }_{C^{\prime },F(C)}\left(1_{F(C)}\circ F(h)\right)\\ & \stackrel{(7.2)}{=}{\Phi }_{C,F(C)}\left(1_{F(C)}\right)\circ h={\eta }_C\circ h,\end{array}$$where we have used (7.1) for $g=1_{F\left(C^{\prime }\right)},k=F(h):F\left(C^{\prime }\right)\to F(C)$ and (7.2) for $g=1_{F(C)}$, $h:C^{\prime }\to C$. Hence the above diagram is commutative, and thus $\eta$ is a natural transformation.

Let $f\in {\mathrm{Hom}}_{\mathcal{C}}(C,U(D))$. Then there is a unique arrow $g\in {\mathrm{Hom}}_{\mathcal{D}}(F(C),D)$ such that $f={\Phi }_{C,D}(g)$. But we have:$$U(g)\circ {\eta }_C=U(g)\circ {\Phi }_{C,F(C)}\left(1_{F(C)}\right)={\Phi }_{C,D}\left(g\circ 1_{F(C)}\right)={\Phi }_{C,D}(g),$$hence $f={\Phi }_{C,D}(g)=U(g)\circ {\eta }_C$. This shows the required universal mapping property for $\eta$.

(2)$⟺$(3) This follows by the duality principle, because we have (1)$⟺$(2).

$\square$

7.3Examples of Adjoints

命题 7.3.1. Adjoints are unique up to an isomorphism, that is, given a functor $F:\mathcal{C}\to$ $\mathcal{D}$ and functors $U,V:\mathcal{D}\to \mathcal{C}$ such that $(F,U)$ and $(F,V)$ are adjoint pairs, then $U\cong V$.

证明. For every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$, we have the following isomorphisms in $\mathbf{Set}$, natural in both $C$ and $D$:$${\mathrm{Hom}}_{\mathcal{C}}(C,U(D))\cong {\mathrm{Hom}}_{\mathcal{D}}(F(C),D)\cong {\mathrm{Hom}}_{\mathcal{C}}(C,V(D))$$Then by the Yoneda lemma, we deduce that $U(D)\cong V(D)$. By adjointness, this isomorphism is again natural in $D$. Hence we have $U\cong V$.

$\square$

例 7.3.2. Consider the following functors:

•

The free functor $F:\mathbf{Set}\to \mathbf{Mon}$ defined by $F(A)=A^{\ast }$ for every set $A$, where $A^{\ast }$ is the free monoid on the set $A$ (i.e. $A^{\ast }$ is the set of “words” with “letters” from $A$, and the operation on $A^{\ast }$ is the concatenation of “words”), and for every function $f:A\to B$, $F(f)=f^{\ast }=\bar{f}:F(A)=A^{\ast }\to F(B)=B^{\ast }$, where $\bar{f}$ is the unique monoid homomorphism with $\bar{f}\circ i_A=i_B\circ f$ and $i_A:A\to A^{\ast }$, $i_B:B\to B^{\ast }$ are the inclusion monoid homomorphisms.

•

The forgetful functor $U:\mathbf{Mon}\to \mathbf{Set}$.

Let us show that $(F,U)$ is an adjunction.

By universal mapping property of the free monoid, for every monoid $(B,\cdot )$ and for every function $f:A\to B$, there is a unique monoid homomorphism $\bar{f}:A^{\ast }\to B$ such that $\bar{f}\circ i_A=f$.

Let $A$ be a set and let $(B,\cdot )$ be a monoid. We prove that there is an isomorphism in Set (bijection):$${\Phi }_{A,B}:{\mathrm{Hom}}_{\mathbf{Mon}}(F(A),B)\to {\mathrm{Hom}}_{\mathbf{Set}}(A,U(B)),$$that is,$${\Phi }_{A,B}:{\mathrm{Hom}}_{\mathbf{Mon}}\left(A^{\ast },B\right)\to {\mathrm{Hom}}_{\mathbf{Set}}(A,B).$$For every $g\in {\mathrm{Hom}}_{\text{Mon }}\left(A^{\ast },B\right)$, we define ${\Phi }_{A,B}(g)=g\circ i_A$. Let us also consider$${\Psi }_{A,B}:{\mathrm{Hom}}_{\mathbf{Set}}(A,B)\to {\mathrm{Hom}}_{\mathbf{Mon}}\left(A^{\ast },B\right)$$defined for every $f\in {\mathrm{Hom}}_{\mathbf{Set}}(A,B)$ by ${\Psi }_{A,B}(f)=\bar{f}$, where $\bar{f}\circ i_A=f$.

Note that $\bar{f}:A^{\ast }\to B$ is defined by $\bar{f}\left(a_1\dots a_n\right)=f\left(a_1\right)\cdots f\left(a_n\right)$.

For every $f\in {\mathrm{Hom}}_{\mathbf{Set}}(A,B)$ we have:$${\Phi }_{A,B}\left({\Psi }_{A,B}(f)\right)={\Phi }_{A,B}(\bar{f})=\bar{f}\circ i_A=f.$$For every $g\in {\mathrm{Hom}}_{\mathbf{Mon}}\left(A^{\ast },B\right)$ we have:$${\Psi }_{A,B}\left({\Phi }_{A,B}(g)\right)={\Psi }_{A,B}\left(g\circ i_A\right)=\overline{g\circ i_A}.$$For every $w=a_1\dots a_n\in A^{\ast }$, we have:$$\overline{g\circ i_A}(w)=\left(g\circ i_A\right)\left(a_1\right)\cdots \left(g\circ i_A\right)\left(a_n\right)=g\left(a_1\right)\cdots g\left(a_n\right)=g(w).$$Hence ${\Psi }_{A,B}\left({\Phi }_{A,B}(g)\right)=g$. Therefore, ${\Phi }_{A,B}$ is a bijection.

We prove that ${\Phi }_{A,B}$ is natural in $A$. Let $\alpha :A\to A^{\prime }$ be a function. Then ${\alpha }^{\ast }:A^{\ast }\to A^{\prime \ast }$. Consider the following diagram:

For every $g\in {\mathrm{Hom}}_{\mathbf{Mon}}\left(A^{\ast },B\right)$, we have:$$\begin{array}{rl}\left[{\Phi }_{A,B}\circ {\mathrm{Hom}}_{\mathbf{Mon}}\left({\alpha }^{\ast },B\right)\right](g)& ={\Phi }_{A,B}\left(g\circ {\alpha }^{\ast }\right)=g\circ {\alpha }^{\ast }\circ i_A=g\circ \bar{\alpha }\circ i_A\\ & =g\circ i_{A^{\prime }}\circ \alpha ={\mathrm{Hom}}_{\mathbf{Set}}(\alpha ,B)\left(g\circ i_{A^{\prime }}\right)\\ & =\left[{\mathrm{Hom}}_{\mathbf{Set}}(\alpha ,B)\circ {\Phi }_{A^{\prime },B}\right](g).\end{array}$$Hence the above diagram is commutative, and thus ${\Phi }_{A,B}$ is natural in $A$.

We prove that ${\Phi }_{A,B}$ is natural in $B$. Let $\beta :B\to B^{\prime }$ be a monoid homomorphism. Consider the following diagram:

For every $g\in {\mathrm{Hom}}_{\mathbf{Mon}}\left(A^{\ast },B\right)$, we have:$$\begin{array}{rl}\left[{\Phi }_{A,B^{\prime }}\circ {\mathrm{Hom}}_{\mathbf{Mon}}\left(A^{\ast },\beta \right)\right](g)& ={\Phi }_{A,B^{\prime }}(\beta \circ g)=\beta \circ g\circ i_A=\beta \circ {\Phi }_{A,B}(g)\\ & =\left[{\mathrm{Hom}}_{\mathbf{Set}}(A,\beta )\circ {\Phi }_{A,B}\right](g).\end{array}$$Hence the above diagram is commutative, and thus ${\Phi }_{A,B}$ is natural in $B$.

Therefore, $(F,U)$ is an adjunction.

Alternatively, one may show that there is the natural transformation $\eta :1_{\mathbf{Set}}\to U\circ F$, whose component at a set $A$ is$${\eta }_A=i_A:A\to (U\circ F)(A)=A^{\ast },$$and prove that it satisfies the required universal mapping property of adjunction. Note that$${\eta }_A={\Phi }_{A,F(A)}\left(1_{F(A)}\right)={\Phi }_{A,F(A)}\left(1_{A^{\ast }}\right)=i_A:A\to A^{\ast }.$$

例 7.3.3. Consider the category $\mathcal{T}$ of all torsion abelian groups (i.e., abelian groups in which every element has finite order) and group homomorphisms. Consider the following functors:

•	The inclusion functor $I:\mathcal{T}\to \mathbf{Ab}$.
•	The torsion functor $T:\mathbf{Ab}\to \mathcal{T}$ defined by $T(G)=t(G)=\{x\in G\mid x\text{has finite order}\}$ for every abelian group $G$, and $T(f)={f|}_{t(G)}:t(G)\to t(H)$ for every group homomorphism $f:G\to H$.

Let us show that $(I,T)$ is an adjunction.

Note that $T(f)={f|}_{t(G)}:t(G)\to t(H)$ is well defined, because if the order of an element $x\in G$ is finite, then the order of $f(x)$ is also finite.

We look for a natural transformation $\eta :1_{\mathcal{T}}\to T\circ I$. For every object $G$ of $\mathcal{T}$, we define$${\eta }_G=1_G:G\to (T\circ I)(G)=t(G)=G.$$Clearly, $\eta$ is a natural transformation. We show universal mapping property for $\eta$. Let $G$ be an object of $\mathcal{T},H$ an object of $\mathbf{Ab}$ and $f:G\to T(H)$ a group homomorphism. We look for a unique group homomorphism $g:I(G)=G\to H$ such that $T(g)\circ {\eta }_G=f$. We have:$$T(g)\circ {\eta }_G=T(g)={g|}_{t(G)}={g|}_G=g.$$In fact, we have $g:G\to t(H)$, and we may take $g=f$.

Therefore, $(I,T)$ is an adjunction.

Alternatively, for every torsion abelian group $A$ and for every abelian group $B$, one proves that there is an isomorphism in $\mathbf{Set}$ (bijection) ${\Phi }_{A,B}:{\mathrm{Hom}}_{\mathbf{Ab}}(I(A),B)\to {\mathrm{Hom}}_{\mathcal{T}}(A,T(B))$, that is, ${\Phi }_{A,B}:{\mathrm{Hom}}_{\mathbf{Ab}}(A,B)\to {\mathrm{Hom}}_{\mathcal{T}}(A,t(B))$. For every $g\in {\mathrm{Hom}}_{\mathbf{Ab}}(I(A),B)={\mathrm{Hom}}_{\mathbf{Ab}}(A,B)$, we have$${\Phi }_{A,B}(g)=T(g)\circ {\eta }_A=T(g)\circ 1_A=T(g)={g|}_{t(A)}={g|}_A=g.$$In fact, we have $g:A\to t(B)$, hence ${\Phi }_{A,B}(g)=g\in {\mathrm{Hom}}_{\mathcal{T}}(A,t(B))$.

例 7.3.4. Let $(A,⩽)$ and $(B,⩽)$ be posets, and let $f:A\to B$ and $g:B\to A$ be order-preserving functions (i.e. functors between poset categories). The following conditions are equivalent:

(i)	For every $a\in A$ and $b\in B$, $f(a)⩽b$ $\iff$ $a⩽g(b)$.
(ii)	For every $a\in A$, $a⩽g(f(a))$, and for every $b\in B$, $f(g(b))⩽b$.

In this case $(f,g)$ is called a monotone Galois connection. In fact, $(f,g)$ is an adjunction between the poset categories $(A,⩽)$ and $(B,⩽)$.

例 7.3.5. Let ${\Omega }^X$ be the set of propositional functions on a set $X$, that is, functions $P:X\to \Omega =\{\perp ,⊺\}$, interpreted as declaring, for each $x\in X$, whether $P(x)$ is false or true. Consider the poset categories $(\Omega ,⩽)$, where $\perp ⩽⊺$, and $\left({\Omega }^X,⩽\right)$, where $P⩽Q$ if and only if $P(x)⩽Q(x)$ for all $x\in X$, which is the case if and only if $P$ implies $Q$. Consider the following functors:

•	${\exists }_X:{\Omega }^X\to \Omega$ defined by ${\exists }_X(P)=⊺$ if and only if there is $x\in X$ with $P(x)=⊺$.
•	The constant “dummy variable” functor ${\Delta }_X:\Omega \to {\Omega }^X$.
•	${\forall }_X:{\Omega }^X\to \Omega$ defined by ${\forall }_X(P)=⊺$ if and only if $P(x)=⊺$ for all $x\in X$.

Then $\left({\exists }_X,{\Delta }_X\right)$ and $\left({\Delta }_X,{\forall }_X\right)$ are adjunctions.

7.4Further Properties of Adjoints

命题 7.4.1. Let $L_1:\mathcal{A}\to \mathcal{B}$, $L_2:\mathcal{B}\to \mathcal{C}$, $R_1:\mathcal{B}\to \mathcal{A}$ and $R_2:\mathcal{C}\to \mathcal{B}$ be functors such that $\left(L_1,R_1\right)$ and $\left(L_2,R_2\right)$ are adjoint pairs. Then $\left(L_2\circ L_1,R_1\circ R_2\right)$ is an adjoint pair.

证明. For every objects $A$ of $\mathcal{A}$ and $C$ of $\mathcal{C}$ we have the natural isomorphism in $\mathbf{Set}$:$$\begin{array}{rl}{\mathrm{Hom}}_{\mathcal{C}}\left(\left(L_2\circ L_1\right)(A),C\right)& \cong {\mathrm{Hom}}_{\mathcal{C}}\left(L_2\left(L_1(A)\right),C\right)\cong {\mathrm{Hom}}_{\mathcal{B}}\left(L_1(A),R_2(C)\right)\\ & \cong {\mathrm{Hom}}_{\mathcal{A}}\left(A,R_1\left(R_2(C)\right)\right)\cong {\mathrm{Hom}}_{\mathcal{A}}\left(A,\left(R_1\circ R_2\right)(C)\right).\end{array}$$Hence $\left(L_2\circ L_1,R_1\circ R_2\right)$ is an adjoint pair.

$\square$

命题 7.4.2. Let $F:\mathcal{C}\to \mathcal{D}$ and $U:\mathcal{D}\to \mathcal{C}$ be functors. The following are equivalent:

(1)

$(F,U)$ is an adjunction.

(2)

There are natural transformations $\eta :1_{\mathcal{C}}\to U\circ F$ and $\epsilon :F\circ U\to 1_{\mathcal{D}}$ such that for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ the following composite functors yield identities on $F(C)$ and $U(D)$ respectively: $$\begin{array}{rl}& F(C)\stackrel{F\left({\eta }_C\right)}{\to }(F\circ U\circ F)(C)\stackrel{{\epsilon }_{F(C)}}{\to }F(C),\\ & U(D)\stackrel{{\eta }_{U(D)}}{\to }(U\circ F\circ U)(D)\stackrel{U\left({\epsilon }_D\right)}{\to }U(D),\end{array}$$that is, ${\epsilon }_{F(C)}\circ F\left({\eta }_C\right)=1_{F(C)}$ and $U\left({\epsilon }_D\right)\circ {\eta }_{U(D)}=1_{U(D)}$. They are shortly written as $\epsilon F\circ F\eta =1_F$ and $U\epsilon \circ \eta U=1_U$, and are called the triangle identities.

证明. (1)$⟹$(2) Assume that $(F,U)$ is an adjunction. By Theorem 7.2.1, for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ there is an isomorphism ${\Phi }_{C,D}:{\mathrm{Hom}}_{\mathcal{D}}(F(C),D)\to {\mathrm{Hom}}_{\mathcal{C}}(C,U(D))$ in $\mathbf{Set}$ with inverse ${\Psi }_{C,D}:{\mathrm{Hom}}_{\mathcal{C}}(C,U(D))\to {\mathrm{Hom}}_{\mathcal{D}}(F(C),D)$.

By Theorem 7.2.1, there exists a natural transformation $\epsilon :F\circ U\to 1_{\mathcal{D}}$ with the following universal mapping property: for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ and for every arrow $g:F(C)\to D$, there is a unique arrow $f:C\to U(D)$ in $\mathcal{C}$ such that ${\epsilon }_D\circ F(f)=g$. In particular, for $D=F(C)$ and $g=1_{F(C)}$, using a relating formula from Theorem 7.2.1, there is a unique arrow $f:C\to U(F(C))$ such that ${\Psi }_{C,F(C)}(f)={\epsilon }_{F(C)}\circ F(f)=1_{F(C)}$. Then$$f={\Phi }_{C,F(C)}\left({\Psi }_{C,F(C)}(f)\right)={\Phi }_{C,F(C)}\left(1_{F(C)}\right)={\eta }_C,$$and thus ${\epsilon }_{F(C)}\circ F\left({\eta }_C\right)=1_{F(C)}$.

By Theorem 7.2.1, there exists a natural transformation $\eta :1_{\mathcal{C}}\to U\circ F$ with the following universal mapping property: for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$ and for every arrow $f:C\to U(D)$, there is a unique arrow $g:F(C)\to D$ in $\mathcal{D}$ such that $U(g)\circ {\eta }_C=f$. In particular, for $C=U(D)$ and $f=1_{U(D)}$, using a relating formula from Theorem 7.2.1, there is a unique arrow $g:F(U(D))\to D$ such that ${\Phi }_{U(D),D}(g)=U(g)\circ {\eta }_{U(D)}=1_{U(D)}$. Then$$g={\Psi }_{U(D),D}\left({\Phi }_{U(D),D}(g)\right)={\Psi }_{U(D),D}\left(1_{U(D)}\right)={\epsilon }_D,$$and thus $U\left({\epsilon }_D\right)\circ {\eta }_{U(D)}=1_{U(D)}$.

(2)$⟹$(1) Suppose that there are natural transformations $\eta :1_{\mathcal{C}}\to U\circ F$ and $\epsilon :F\circ U\to 1_{\mathcal{D}}$ such that for every objects $C$ of $\mathcal{C}$ and $D$ of $\mathcal{D}$, we have ${\epsilon }_{F(C)}\circ F\left({\eta }_C\right)=1_{F(C)}$ and $U\left({\epsilon }_D\right)\circ {\eta }_{U(D)}=1_{U(D)}$.

We show that $\eta$ has universal mapping property from Theorem 7.2.1. Let $C$ be an object of $\mathcal{C}$, let $D$ be an object of $\mathcal{D}$ and let $f:C\to U(D)$ be an arrow in $\mathcal{C}$. We look for a unique arrow $g:F(C)\to D$ such that $U(g)\circ {\eta }_C=f$. Choose $g={\epsilon }_D\circ F(f)$. By the naturality of $\eta$, we have the following commutative diagram:

hence $(U\circ F)(f)\circ {\eta }_C={\eta }_{U(D)}\circ f$. It follows that:$$U(g)\circ {\eta }_C=U\left({\epsilon }_D\right)\circ (U\circ F)(f)\circ {\eta }_C=U\left({\epsilon }_D\right)\circ {\eta }_{U(D)}\circ f=1_{U(D)}\circ f=f.$$

For uniqueness, suppose that there is also an arrow $g^{\prime }:F(C)\to D$ such that $U\left(g^{\prime }\right)\circ {\eta }_C=f=U(g)\circ {\eta }_C$. Then $F\left(U\left(g^{\prime }\right)\right)\circ F\left({\eta }_C\right)=F(U(g))\circ F\left({\eta }_C\right)$. By the naturality of $\epsilon$, we have the following commutative diagram:

hence $g\circ {\epsilon }_{F(C)}={\epsilon }_D\circ (F\circ U)(g)$, and a similar equality for $g^{\prime }$. It follows that:$$\begin{array}{rl}{g}^{\prime }& =g^{\prime }\circ 1_{F(C)}=g^{\prime }\circ {\epsilon }_{F(C)}\circ F\left({\eta }_C\right)={\epsilon }_D\circ (F\circ U)\left(g^{\prime }\right)\circ F\left({\eta }_C\right)\\ & ={\epsilon }_D\circ F\left(U\left(g^{\prime }\right)\circ {\eta }_C\right)={\epsilon }_D\circ F\left(U(g)\circ {\eta }_C\right)=g\circ {\epsilon }_{F(C)}\circ F\left({\eta }_C\right)=g\circ 1_{F(C)}=g.\end{array}$$Therefore, $(F,U)$ is an adjunction by Theorem 7.2.1.

$\square$

推论 7.4.3. Let $(L,R)$ be an adjunction with $L:\mathcal{A}\to \mathcal{B}$, $R:\mathcal{B}\to \mathcal{A}$, unit $\eta :1_{\mathcal{A}}\to R\circ L$ and counit $\epsilon :L\circ R\to 1_{\mathcal{B}}$. Consider the following full subcategory of $\mathcal{A}$:$$\mathrm{Fix}(R\circ L)=\left\{A\in \mathcal{A}\mid {\eta }_A:A\to R(L(A))\text{is an isomorphism}\right\},$$and the following full subcategory of $\mathcal{B}$:$$\mathrm{Fix}(L\circ R)=\left\{B\in \mathcal{B}\mid {\epsilon }_B:L(R(B))\to B\text{is an isomorphism}\right\}$$Then $(L,R)$ restricts to an equivalence between the categories $\mathrm{Fix}(R\circ L)$ and $\mathrm{Fix}(L\circ R)$.

证明. Let $A$ be an object of $\mathrm{Fix}(R\circ L)$. Then ${\eta }_A:A\to R(L(A))$ is an isomorphism. It follows that $L\left({\eta }_A\right)$ is an isomorphism. By the previous proposition we have ${\epsilon }_{L(A)}\circ L\left({\eta }_A\right)=1_{L(A)}$. It follows that ${\epsilon }_{L(A)}:L(R(L(A)))\to L(A)$ is an isomorphism. This shows that $L(A)\in \mathrm{Fix}(L\circ R)$. Hence we may consider the functor $L^{\prime }:\mathrm{Fix}(R\circ L)\to \mathrm{Fix}(L\circ R)$, which is the restriction of the functor $L$ to $\mathrm{Fix}(R\circ L)$.

Also, let $B$ be an object of $\mathrm{Fix}(L\circ R)$. Then ${\epsilon }_B:L(R(B))\to B$ is an isomorphism. It follows that $R\left({\epsilon }_B\right)$ is an isomorphism. By the previous proposition we have $R\left({\epsilon }_B\right)\circ {\eta }_{R(B)}=1_{R(B)}$. It follows that ${\eta }_{R(B)}:R(B)\to R(L(R(B)))$ is an isomorphism. This shows that $R(B)\in \mathrm{Fix}(R\circ L)$. Hence we may consider the functor $R^{\prime }:\mathrm{Fix}(L\circ R)\to \mathrm{Fix}(R\circ L)$, which is the restriction of the functor $R$ to $\mathrm{Fix}(L\circ R)$.

Clearly, the restrictions ${\eta }^{\prime }:1_{\mathrm{Fix}(R\circ L)}\to R^{\prime }\circ L^{\prime }$ of $\eta$ and ${\epsilon }^{\prime }:L^{\prime }\circ R^{\prime }\to 1_{\mathrm{Fix}(L\circ R)}$ of $\epsilon$ are natural isomorphisms. Hence $L^{\prime }$ and $R^{\prime }$ define an equivalence between the categories $\mathrm{Fix}(R\circ L)$ and $\mathrm{Fix}(L\circ R)$.

$\square$

例 7.4.4. An equivalence of categories with associated natural isomorphisms $\eta$ and $\epsilon$ need not be an adjunction with unit $\eta$ and counit $\epsilon$. For instance, let $(M,\cdot )$ and $(N,\cdot )$ be monoids, and view them as monoid categories. An equivalence of categories between them is a pair of functors, that is, monoid homomorphisms $f:M\to N$ and $g:N\to M$ and two natural isomorphisms $\alpha :1_M\to g\circ f$ and $\beta :f\circ g\to 1_N$, that is, invertible elements $\alpha \in M$ and $\beta \in N$. The triangle identities imply $f(\alpha )=\beta$ and $g(\beta )=\alpha$. Taking $M=N$ and $f=g=1_M$, an equivalence of categories satisfying the triangle identities consists of two central invertible elements $\alpha ,\beta \in M$ such that $\alpha =\beta$. Now let $(M,\cdot )$ be a non-trivial group and $\alpha ,\beta \in M$ be distinct. Then $(\alpha ,\beta )$ is an equivalence which is not an adjunction.

命题 7.4.5. If $(F,U)$ is an equivalence of categories with $F:\mathcal{C}\to \mathcal{D}$, $U:\mathcal{D}\to \mathcal{C}$ and natural isomorphisms $\eta :1_{\mathcal{C}}\to U\circ F$ and $\epsilon :F\circ U\to 1_{\mathcal{D}}$, then there is a unique natural isomorphism ${\eta }_0:1_{\mathcal{C}}\to U\circ F$ such that $(F,U)$ is an adjoint pair with unit ${\eta }_0$ and counit $\epsilon$, and a unique natural isomorphism ${\epsilon }_0:F\circ U\to 1_{\mathcal{D}}$ such that $(F,U)$ is an adjoint pair with unit $\eta$ and counit ${\epsilon }_0$.

We say that every equivalence of categories can be turned into an adjoint equivalence.

命题 7.4.6 (RAPL, LAPC). Right adjoints preserve limits and left adjoints preserve colimits. In particular, right adjoints preserve products, terminal objects, equalizers and pullbacks, while left adjoints preserve coproducts, initial objects, coequalizers and pushouts.

证明. Let $F:\mathcal{C}\to \mathcal{D}$ and $U:\mathcal{D}\to \mathcal{C}$ be functors such that $(F,U)$ is an adjoint pair. Let $D:\mathcal{J}\to \mathcal{D}$ be a diagram such that there is $\underset{⟵}{\lim }{D}_j$ in $\mathcal{D}$. For every object $X$ of $\mathcal{C}$ we have the natural isomorphisms in $\mathbf{Set}$:$$\begin{array}{rl}{\mathrm{Hom}}_{\mathcal{C}}(X,U(\underset{⟵}{\lim }{D}_j))& \cong {\mathrm{Hom}}_{\mathcal{D}}(F(X),\underset{⟵}{\lim }{D}_j)\cong \underset{⟵}{\lim }{\mathrm{Hom}}_{\mathcal{D}}\left(F(X),D_j\right)\\ & \cong \underset{⟵}{\lim }{\mathrm{Hom}}_{\mathcal{C}}(X,U(D_j))\cong {\mathrm{Hom}}_{\mathcal{C}}(X,\underset{⟵}{\lim }U(D_j)),\end{array}$$where we have used the adjunction isomorphisms and the fact that the functor ${\mathrm{Hom}}_{\mathcal{D}}(F(C),-)$ preserves limits. The by the Yoneda lemma, we deduce that $U(\underset{⟵}{\lim }{D}_j)\cong \underset{⟵}{\lim }U(D_j)$.

$\square$

例 7.4.7. (1) The above property may be useful in practice to show that a given functor does not have a left or right adjoint.

First, consider the inclusion functor $I:\mathbf{Ring}\to \mathbf{Rng}$. If $I$ has a right adjoint, then $I$ is a left adjoint and needs to preserve colimits, and in particular, initial objects. But the initial object of $\mathbf{Ring}$ is $\mathbb{Z}$, while the initial object of $\mathbf{Rng}$ is $\{0\}$. Hence $I$ does not have a right adjoint. One may show that it has a left adjoint given by the unitary (Dorroh) extension of a ring.

Now consider the inclusion functor $I:\mathcal{T}\to \mathbf{Ab}$. We have seen that it has the torsion functor as a right adjoint. If $I$ has a left adjoint, then it is a right adjoint and needs to preserve limits, and in particular, products. But the product in $\mathcal{T}$ of the family of torsion abelian groups ${\left({\mathbb{Z}}_p\right)}_{p\text{prime}}$ is $t\left({\prod }_{p\text{prime}}{\mathbb{Z}}_p\right)={⨁}_{p\text{prime}}{\mathbb{Z}}_p$, while its product in $\mathbf{Ab}$ is their direct product. Hence $T$ does not have a left adjoint.

(2) There are functors without any left or right adjoint. For instance take the functor $F:0\to \mathcal{A}$ from the zero category to a non-zero category $\mathcal{A}$.

Also, there are functors having both a left and a right adjoint. E.g., see the example with the logical quantifiers. Alternatively, consider the following functors:

•	The functor $L:\mathbf{Set}\to \mathbf{Top}$ defined by $L(X)=(X,\mathcal{P}(X))$ (discrete topology), and $L(f)=f:(X,\mathcal{P}(X))\to (X,\mathcal{P}(Y))$ for every function $f:X\to Y$.
•	The forgetful functor $U:\mathbf{Top}\to \mathbf{Set}$.
•	The functor $R:\mathbf{Set}\to \mathbf{Top}$ defined by $R(X)=(X,\{\varnothing ,X\})$ (indiscrete topology), and $R(f)=f:(X,\{\varnothing ,X\})\to (Y,\{\varnothing ,Y\})$ for every function $f:X\to Y$.

Then $(L,U)$ and $(U,R)$ are adjunctions.

注 7.4.8. There are some general results, such as Freyd’s Adjoint Functor Theorem, on the existence of left or right adjoints.