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6.1.2 Adjuncts

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories. By virtue of Exercise 6.1.0.4, every adjunction $(\eta , \epsilon )$ between $F$ and $G$ determines a collection of bijections

\[ \rho _{C,D}: \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) ), \]

depending functorially on $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$. In this section, we establish an analogue of this statement for adjunctions in an arbitrary $2$-category.

Construction 6.1.2.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $T$, $C$, and $D$, together with $1$-morphisms $f: C \rightarrow D$, $g: D \rightarrow C$, $c: T \rightarrow C$, and $d: T \rightarrow D$.

  • Let $\epsilon : f \circ g \Rightarrow \operatorname{id}_ D$ and $\beta : x \Rightarrow g \circ d$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. We will refer to the composition

    \[ f \circ c \xRightarrow {\operatorname{id}_ f \circ \beta } f \circ (g \circ d) \xRightarrow [\sim ]{\alpha _{f,g,d}} (f \circ g) \circ d \xRightarrow {\epsilon \circ \operatorname{id}_ d} \operatorname{id}_ D \circ d \xRightarrow [\sim ]{\lambda _ d} d \]

    as the left adjunct of $\beta $ with respect to $\epsilon $, or more simply as the left adjunct of $\beta $ if the $2$-morphism $\epsilon $ is clear from context. Here $\lambda _{d}$ and $\alpha _{f,g,d}$ are the left unit and associativity constraints for the $2$-category $\operatorname{\mathcal{C}}$.

  • Let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\gamma : f \circ c \Rightarrow d$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. We will refer to the composition

    \[ c \xRightarrow [\sim ]{ \lambda _{c}^{-1} } \operatorname{id}_{C} \circ c \xRightarrow {\eta } (g \circ f) \circ c \xRightarrow [\sim ]{ \alpha _{g,f,c}^{-1} } g \circ (f \circ c) \xRightarrow {\operatorname{id}_ g \circ \gamma } g \circ d \]

    as the right adjunct of $\gamma $ with respect to $\eta $, or more simply as the right adjunct of $\gamma $ if the $2$-morphism $\eta $ is clear from context. Here again $\lambda _{c}$ and $\alpha _{g,f,c}$ are the left unit and associativity constraints for the $2$-category $\operatorname{\mathcal{C}}$.

Example 6.1.2.2. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing $1$-morphisms $f: C \rightarrow D$ and $g: D \rightarrow C$. Then:

  • Every $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is equal to the right adjunct of the right unit constraint $\rho _{f}: f \circ \operatorname{id}_{D} \xRightarrow {\sim } f$ (with respect to $\eta $).

  • Every $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ is equal to the left adjunct of $\rho _{g}^{-1}: g \xRightarrow {\sim } g \circ \operatorname{id}_{D}$ (with respect to $\epsilon $).

Example 6.1.2.3. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing $1$-morphisms $f: C \rightarrow D$ and $g: D \rightarrow C$, and suppose we are given $2$-morphisms $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$. Then $(\eta , \epsilon )$ is an adjunction between $f$ and $g$ if and only if the following conditions are satisfied:

$(Z1)$

The left adjunct of $\eta $ (with respect to $\epsilon $) is equal to the right unit constraint $\rho _{f}: f \circ \operatorname{id}_{C} \xRightarrow {\sim } f$.

$(Z2)$

The right adjunct of $\epsilon $ (with respect to $\eta $) is the inverse $\rho _{g}^{-1}: g \xRightarrow {\sim } g \circ \operatorname{id}_{D}$ of the right unit constraint.

Remark 6.1.2.4 (Functoriality). Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $T$, $C$, and $D$, together with $1$-morphisms $f: C \rightarrow D$, $g: D \rightarrow C$, $c,c': T \rightarrow C$, and $d,d': T \rightarrow D$. Then:

  • If $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\varphi : c \Rightarrow c'$ are $2$-morphisms of $\operatorname{\mathcal{C}}$, then the diagram of sets

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c', d) \ar [r] \ar [d]^{ \operatorname{id}_ f \circ \varphi } & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c', g \circ d) \ar [d]^{ \varphi } \\ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c, d) \ar [r] & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c, g \circ d) } \]

    is commutative, where the horizontal maps are given by the formation of right adjuncts with respect to $\eta $.

  • If $\epsilon : f \circ g \Rightarrow \operatorname{id}_ D$ and $\varphi : c \Rightarrow c'$ are $2$-morphisms of $\operatorname{\mathcal{C}}$, then the diagram of sets

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c', g \circ d) \ar [d]^{ \varphi } \ar [r] & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c', d) \ar [d]^{ \operatorname{id}_ f \circ \varphi } & \\ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c, g \circ d) \ar [r] & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c, d), } \]

    where the horizontal maps are given by the formation of left adjuncts with respect to $\epsilon $.

  • If $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\psi : d \Rightarrow d'$ are $2$-morphisms of $\operatorname{\mathcal{C}}$, then the diagram of sets

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c, d) \ar [r] \ar [d]^{ \psi } & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c, g \circ d) \ar [d]^{ \operatorname{id}_ g \circ \psi } \\ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c, d') \ar [r] & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c, g \circ d') } \]

    is commutative, where the horizontal maps are given by the formation of right adjuncts with respect to $\eta $.

  • If $\epsilon : f \circ g \Rightarrow \operatorname{id}_ D$ and $\psi : d \Rightarrow d'$ are $2$-morphisms of $\operatorname{\mathcal{C}}$, then the diagram of sets

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c, g \circ d) \ar [d]^{ \operatorname{id}_ g \circ \psi } \ar [r] & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c, d) \ar [d]^{ \psi } & \\ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, C)}( c, g \circ d') \ar [r] & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D)}( f \circ c, d'), } \]

    is commutative, where the horizontal maps are given by the formation of left adjuncts with respect to $\epsilon $.

Stated more informally, Construction 6.1.2.1 depends functorially on the $1$-morphisms $c: T \rightarrow C$ and $d: T \rightarrow D$.

Proposition 6.1.2.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ be $2$-morphisms. Suppose we are given another object $T \in \operatorname{\mathcal{C}}$ equipped with $1$-morphisms $c: T \rightarrow C$ and $d: T \rightarrow D$, together with $2$-morphisms $\beta : c \Rightarrow g \circ d$ and $\gamma : f \circ c \Rightarrow d$. Then:

$(1)$

If the pair $(\eta , \epsilon )$ satisfies condition $(Z1)$ of Definition 6.1.1.1 and $\beta $ is the right adjunct of $\gamma $, then $\gamma $ is the left adjunct of $\beta $.

$(2)$

If the pair $(\eta , \epsilon )$ satisfies condition $(Z2)$ of Definition 6.1.1.1 and $\gamma $ is the left adjunct of $\beta $, then $\beta $ is the right adjunct of $\gamma $.

Proof. We will prove $(1)$; the proof of $(2)$ follows by applying the same argument in the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$. Consider the diagram

\[ \xymatrix@C =15pt@R=50pt{ f \circ c \ar@ {=>}[r]^-{ \lambda _{c}^{-1} } \ar@ {=>}[d]^{\rho _{f}^{-1}} & f \circ (\operatorname{id}_{C} \circ c) \ar@ {=>}[dl]^{\sim } \ar@ {=>}[r]^-{\eta } & f \circ ((g \circ f) \circ c) \ar@ {=>}[r]^-{\sim } \ar@ {=>}[dl]_{\sim } & f \circ (g \circ (f \circ c))) \ar@ {=>}[r]^-{\gamma } \ar@ {=>}[d]_{\sim } & f \circ (g \circ d) \ar@ {=>}[d]_{\sim } \\ ( f \circ \operatorname{id}_ C) \circ c \ar@ {=>}[r]^-{\eta } & (f \circ (g \circ f) \circ c \ar@ {=>}[r]^-{\sim } & ((f \circ g) \circ f) \circ c \ar@ {=>}[r]^-{\sim } \ar@ {=>}[d]^{\epsilon } & (f \circ g) \circ (f \circ c) \ar@ {=>}[d]^{\epsilon } \ar@ {=>}[r]^-{\gamma } & (f \circ g) \circ c \ar@ {=>}[d]^{\epsilon } \\ & & (\operatorname{id}_ D \circ f) \circ c \ar@ {=>}[r]^-{\sim } \ar@ {=>}[dr]^-{\lambda _{f}} & \operatorname{id}_ D \circ (f \circ c) \ar@ {=>}[r]^-{\gamma } \ar@ {=>}[d]^{\lambda _{f \circ c}} & \operatorname{id}_ D \circ d \ar@ {=>}[d]^{\lambda _ d} \\ & & & f \circ c \ar@ {=>}[r]^-{\gamma } & d } \]

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)$, where the unlabeled morphisms are given by the associativity constraints of $\operatorname{\mathcal{C}}$ (and their inverses). Our assumption that $\beta $ is the right adjunct of $\gamma $ guarantees that the composition along the top line coincides with $\operatorname{id}_{f} \circ \beta $. Consequently, the left adjunct of $\beta $ is the $2$-morphism of $\operatorname{\mathcal{C}}$ given by clockwise composition around the outside of the diagram. On the other hand, axiom $(Z1)$ of Definition 6.1.1.1 guarantees counterclockwise composition around the outside of the diagram coincides with $\gamma $. To complete the proof, it will suffice to show that the diagram commutes. The commutativity of the triangular regions follows Propositions 2.2.1.14 and 2.2.1.16. The commutativity of the bottom right square follows from the naturality of left unit constraints (Remark 2.2.1.13) and the commutativity of the middle right square from the functoriality of composition. The remaining squares commute by the naturality of the associativity constraints of $\operatorname{\mathcal{C}}$, and the five-sided region commutes by virtue of the pentagon identity. $\square$

Corollary 6.1.2.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and suppose we are given $2$-morphisms $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$. The following conditions are equivalent:

$(1)$

The pair $(\eta , \epsilon )$ is an adjunction between $f$ and $g$ (in the sense of Definition 6.1.1.1).

$(2)$

For every object $T \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $c: T \rightarrow C$ and $d: T \rightarrow D$, the formation of left and right adjuncts (Construction 6.1.2.1) supplies mutually inverse bijections

\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)}( f \circ c, d ) \simeq \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,C)}( c, f \circ d ). \]

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 6.1.2.5. For the converse, we first observe that $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is equal to the right adjunct of the right unit constraint $\rho _{f}: f \circ \operatorname{id}_{D} \xRightarrow {\sim } f$ with respect to $\eta $ (Example 6.1.2.2). If assumption $(2)$ is satisfied, then $\rho _{f}$ is the left adjunct of $\eta $ with respect to $\epsilon $. Similarly, assumption $(2)$ guarantees that $\rho _{g}^{-1}: g \xRightarrow {\sim } g \circ \operatorname{id}_{D}$ is the right adjunct of $\epsilon $ with respect to $\eta $, so that the pair $(\eta , \epsilon )$ is an adjunction by virtue of Example 6.1.2.3. $\square$

Example 6.1.2.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories, and let $(\eta , \epsilon )$ be an adjunction between $F$ and $G$. Suppose we are given objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, which we identify with functors $C: \{ \ast \} \rightarrow \operatorname{\mathcal{C}}$ and $D: \{ \ast \} \rightarrow \operatorname{\mathcal{D}}$, respectively. Applying Corollary 6.1.2.6 to the $2$-category $\mathbf{Cat}$, we obtain a bijection

\[ \rho _{C,D}: \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{D}}}( C, G(D) ). \]

This bijection depends functorially on $C$ and $D$ (Remark 6.1.2.4), and can therefore be regarded as a $\operatorname{Hom}$-adjunction between $F$ and $G$ in the sense of Definition 6.1.0.1. Note that, for every morphism $f: F(C) \rightarrow D$ in $\operatorname{\mathcal{C}}$, the image $\rho _{C,D}(f) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) )$ is given explicitly by the composition $C \xrightarrow { \eta _ C } (G \circ F)(C) \xrightarrow { G(f)} G(D)$. In particular, the morphism $\eta _ C: C \rightarrow (G \circ F)(C)$ can be recovered by applying $\rho _{C,F(C)}$ to the identity morphism $\operatorname{id}_{ F(C) }$. Similarly, for each object $D \in \operatorname{\mathcal{D}}$, the morphism $\epsilon _ D: (F \circ G)(D) \rightarrow D$ can be recovered by applying $\rho _{G(D), D}^{-1}$ to the identity morphism $\operatorname{id}_{G(D)}$. In other words, the adjunction $(\eta , \epsilon )$ is obtained by applying the construction of Example 6.1.0.3 to the $\operatorname{Hom}$-adjunction $\{ \rho _{C,D} \} _{C \in \operatorname{\mathcal{C}}, D \in \operatorname{\mathcal{D}}}$.

Corollary 6.1.2.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and suppose we are given $2$-morphisms $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ satisfying condition $(Z1)$ of Definition 6.1.1.1. Let $\gamma : g \Rightarrow g$ denote the $2$-morphism given by the composition

\[ g \xRightarrow [\sim ]{\lambda _{g}^{-1}} \operatorname{id}_{C} \circ g \xRightarrow {\eta \circ \operatorname{id}_ g} (g \circ f) \circ g \xRightarrow [\sim ]{\alpha _{g,f,g}^{-1}} g \circ (f \circ g) \xRightarrow {\operatorname{id}_{g} \circ \epsilon } g \circ \operatorname{id}_ D \xRightarrow [\sim ]{\rho _{g}} g. \]

Then $\gamma $ is idempotent: that is, $\gamma ^2 = \gamma $ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)$. In particular, if $\gamma $ has either a left or a right inverse, then $\gamma = \operatorname{id}_{g}$ (so that $(\eta , \epsilon )$ is an adjunction between $f$ and $g$).

Proof. Let $\gamma '$ denote the composition $g \xRightarrow {\gamma } g \xRightarrow { \rho _{g}^{-1} } g \circ \operatorname{id}_ D$. Then $\gamma '$ is the right adjunct of $\epsilon $ with respect to $\eta $ (see Example 6.1.2.3). Invoking Remark 6.1.2.4, we deduce that the horizontal composition $\gamma ' \gamma $ is the right adjunct of $\epsilon '$ with respect to $\eta $, where $\epsilon '$ denotes the composite map $f \circ g \xRightarrow { \operatorname{id}_ f \circ \gamma } f \circ g \xRightarrow {\epsilon } \operatorname{id}_ D$. Combining Example 6.1.2.2 with Remark 6.1.2.4, we see that $\epsilon '$ is the left adjunct of $\gamma '$ with respect to $\epsilon $. Since the pair $(\eta , \epsilon )$ satisfies $(Z1)$, it follows that $\gamma ' \gamma = \gamma '$, Composing with the right unit constraint $\rho _ g$, we conclude that $\gamma \gamma = \gamma $. $\square$

Proposition 6.1.2.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

For every object $T \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $c: T \rightarrow C$ and $d: T \rightarrow D$, the formation of right adjuncts with respect to $\eta $ (Construction 6.1.2.1) induces a bijection

\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)}( f \circ c, d ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,C)}( c, g \circ d). \]
$(2)$

There exists a $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ for which $(\eta , \epsilon )$ is an adjunction between $f$ and $g$.

Moreover, if these conditions are satisfied, then the $2$-morphism $\epsilon $ is uniquely determined.

Proof. The implication $(2) \Rightarrow (1)$ follows from Corollary 6.1.2.6. Conversely, suppose that condition $(1)$ is satisfied. Applying $(1)$ in the case $T = D$, $c = g$, and $d = \operatorname{id}_{D}$, we conclude that there is a unique $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ whose right adjunct is equal to the inverse $\rho _{g}^{-1}: g \xRightarrow {\sim } g \circ \operatorname{id}_{D}$ of the right unit constraint $\rho _ g$, so that the pair $(\eta , \epsilon )$ satisfies condition $(Z2)$ of Definition 6.1.1.1 (Example 6.1.2.3). We will complete the proof by showing that $(\eta , \epsilon )$ also satisfies condition $(Z1)$. Let $\gamma : f \circ \operatorname{id}_{C} \Rightarrow f$ be the left adjunct of $\eta $. It follows from Proposition 6.1.2.5 that the right adjunct of $\gamma $ is equal to $\eta $, which is also the right adjunct of the unit constraint $\rho _{f}: f \circ \operatorname{id}_{C} \xRightarrow {\sim } f$. Invoking assumption $(1)$, we conclude that $\gamma = \rho _{f}$, which is a restatement of $(Z1)$ (Example 6.1.2.3). $\square$

Definition 6.1.2.10. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$. We say that a $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is the unit of an adjunction if it satisfies the equivalent conditions of Proposition 6.1.2.9: that is, if there exists a $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ for which the pair $(\eta , \epsilon )$ is an adjunction. If this condition is satisfied, we will say that $\eta $ exhibits $f$ as a left adjoint of $g$ and also that $\eta $ exhibits $g$ as a right adjoint of $f$.

In the $2$-category $\mathbf{Cat}$, we can formulate a sharper version of Proposition 6.1.2.9:

Variant 6.1.2.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. The following conditions are equivalent:

$(1)$

For every pair of objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, the formation of right adjuncts with respect to $\eta $ induces a bijection $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) )$.

$(2)$

There exists a natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ for which $(\eta , \epsilon )$ is an adjunction between $F$ and $G$.

Moreover, if these conditions are satisfied, then the natural transformation $\epsilon $ is uniquely determined.

Proof. We will prove that $(1) \Rightarrow (2)$; the remaining assertions follow immediately from Proposition 6.1.2.9. Fix an object $D \in \operatorname{\mathcal{D}}$. Applying assertion $(1)$ in the case $C = G(D)$, we deduce that there is a unique morphism $\epsilon _{D}: (F \circ G)(D) \rightarrow D$ for which the composition

\[ G(D) \xrightarrow {\eta _{G(D)}} (G \circ F \circ G)(D) \xrightarrow { G( \epsilon _ D)} G(D) \]

is the identity morphism from $G(D)$ to itself.

We first claim that the construction $D \mapsto \epsilon _ D$ is a natural transformation of functors from $F \circ G$ to $\operatorname{id}_{\operatorname{\mathcal{D}}}$. Let $h: D \rightarrow D'$ be a morphism in the category $\operatorname{\mathcal{D}}$; we wish to show that the diagram

\[ \xymatrix@R =50pt@C=50pt{ (F \circ G)(D) \ar [d]^-{ (F \circ G)(h) } \ar [r]^-{ \epsilon _ D } & D \ar [d]^{h} \\ (F \circ G)(D') \ar [r]^-{ \epsilon _{D'} } & D' } \]

commutes. Consider the diagram

\[ \xymatrix@R =50pt@C=50pt{ G(D) \ar [r]^-{ \eta _{G(D)} } \ar [d]^{ G(h) } & (G \circ F \circ G)(D) \ar [r]^-{ G(\epsilon _ D)} \ar [d]^{G(F(G(h))) } & G(D) \ar [d]^{ G(h) } \ar [d] \\ G(D') \ar [r]^-{ \eta _{G(D')} } & (G \circ F \circ G)(D') \ar [r]^-{ G(\epsilon _{D'}) } & G(D') } \]

in the category $\operatorname{\mathcal{C}}$. It follows from the definitions of $\epsilon _ D$ and $\epsilon _{D'}$ that both horizontal compositions are equal to the identity, so the outer rectangle commutes. Since $\eta $ is a natural transformation, the left square commutes. It follows that the compositions $G(h) \circ G(\epsilon _ D) \circ u_{G(D)}$ and $G(\epsilon _{D'}) \circ G(F(G(h))) \circ \eta _{G(D)}$ are the same: that is, the morphisms

\[ h \circ \epsilon _ D, \epsilon _{D'} \circ F(G(h)) \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (F \circ G)(D), D') \]

have the same right adjunct. Invoking assumption $(1)$, we deduce that $h \circ \epsilon _{D} = \epsilon _{D'} \circ (F(G(h))$, as desired.

It follows immediately from the construction that the pair of natural transformations $(\eta , \epsilon )$ satisfies condition $(Z2)$ of Definition 6.1.0.2. To complete the proof, it will suffice to show that it also satisfies condition $(Z1)$. Let $C$ be an object of $\operatorname{\mathcal{C}}$; we wish to show that the composite map

\[ F(C) \xrightarrow { F( \eta _ C )} (F \circ G \circ F)(C) \xrightarrow { \epsilon _{ F(C) } } F(C) \]

is equal to the identity map $\operatorname{id}_{F(C)}$. Note that the right adjunct of $\epsilon _{F(C)} \circ F(\eta _ C)$ is the composite map

\[ C \xrightarrow {\eta _ C} (G \circ F)(C) \xrightarrow { (G \circ F)(\eta _ C) } (G \circ F \circ G \circ F)(C) \xrightarrow { G( \epsilon _{F(C)})} (G \circ F)(C). \]

By virtue of the fact that $(\eta , \epsilon )$ satisfies $(Z2)$, this composition is equal to $\eta _ C$, which is also the right adjunct of the identity map $\operatorname{id}_{F(C)}$. Invoking assumption $(1)$, we conclude that $\epsilon _{F(C)} \circ F(\eta _ C) = \operatorname{id}_{ F(C)}$, as desired. $\square$

We can give another characterization of the units of adjunctions by applying Proposition 6.1.2.9 in the opposite $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$:

Variant 6.1.2.12. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: C \rightarrow D$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. Then $\eta $ is the unit of an adjunction if and only if the following condition is satisfied:

  • For every object $T \in \operatorname{\mathcal{C}}$ and every pair of morphisms $c: C \rightarrow T$ and $d: D \rightarrow T$, the $2$-morphism $\eta $ determines a bijection

    \[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,T)}( c \circ g, d ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,T)}( c, d \circ f ), \]

    carrying each $2$-morphism $\beta : c \circ g \Rightarrow d$ to the composition

    \[ c \xRightarrow [\sim ]{\rho _{c}^{-1}} c \circ \operatorname{id}_{C} \xRightarrow {\operatorname{id}_ c \circ \eta } c \circ (g \circ f) \xRightarrow [\sim ]{\alpha _{c,g,f}} (c \circ g) \circ f \xRightarrow {\beta \circ \operatorname{id}_ f} d \circ f. \]

For the reader's convenience, let us also record a conjugate version of the preceding discussion:

Proposition 6.1.2.13. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

For every object $T \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $c: T \rightarrow C$ and $d: T \rightarrow D$, the formation of left adjuncts with respect to $\epsilon $ (Construction 6.1.2.1) induces a bijection

\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,C)}( c, g \circ d) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)}( f \circ c, d ) \]
$(2)$

For every object $T \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $c: C \rightarrow T$ and $d: D \rightarrow T$, the $2$-morphism $\epsilon $ determines a bijection

\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,T)}( c, d \circ f ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,T)}( c \circ g, d ) \]

carrying each $2$-morphism $\gamma : c \Rightarrow d \circ f$ to the composition

\[ c \circ g \xRightarrow {\gamma \circ \operatorname{id}_ g} (d \circ f) \circ g \xRightarrow [\sim ]{\alpha _{d,f,g}^{-1}} d \circ (f \circ g) \xRightarrow {\operatorname{id}_{d} \circ \epsilon } d \circ \operatorname{id}_{D} \xRightarrow [\sim ]{\rho _{d}} d. \]
$(3)$

There exists a $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ for which $(\eta , \epsilon )$ is an adjunction between $f$ and $g$.

Moreover, if these conditions are satisfied, then the $2$-morphism $\eta $ is uniquely determined.

Proof. Apply Proposition 6.1.2.9 and Variant 6.1.2.12 to the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$. $\square$

Definition 6.1.2.14. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$. We say that a $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ is the counit of an adjunction if it satisfies the equivalent conditions of Proposition 6.1.2.13: that is, there exists a $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ for which the pair $(\eta , \epsilon )$ is an adjunction. If this condition is satisfied, we will say that $\epsilon $ exhibits $f$ as a left adjoint of $g$ and also that $\epsilon $ exhibits $g$ as a right adjoint of $f$.