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Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ be a $1$-morphism of $\operatorname{\mathcal{C}}$. We will say that a $1$-morphism $g: D \rightarrow C$ is a right adjoint of $f$ if there exists an adjunction $(\eta , \epsilon )$ between $f$ and $g$, in the sense of Definition 6.1.1.1. Beware that the right adjoint of $f$ is usually not unique: if $g$ is a right adjoint of $f$, then any $1$-morphism $g': D \rightarrow C$ which is isomorphic to $g$ can also be regarded as a right adjoint to $f$ (see Remark 6.1.1.5). However, we will show in this section that this is the only source of ambiguity: the right adjoint of a $1$-morphism $f$ (if it exists) is well-defined up to canonical isomorphism.

Proposition 6.1.3.1. 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 the unit of an adjunction. Then:

$(1)$

For every $1$-morphism $f': C \rightarrow D$, the function

$\operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)}( f, f' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_ C, g \circ f') \quad \quad \gamma \mapsto (\operatorname{id}_ g \circ \gamma )\eta$

is a bijection.

$(2)$

For every $1$-morphism $g': D \rightarrow C$, the function

$\operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)}( g, g' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_ C, g' \circ f) \quad \quad \beta \mapsto (\beta \circ \operatorname{id}_{f}) \eta$

is a bijection.

Proof. Let $\rho _{f}: f \circ \operatorname{id}_{C} \xRightarrow {\sim } f$ be the right unit constraint. To prove $(1)$, we observe that the composition

$\operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)}( f \circ \operatorname{id}_ C, f' ) \xrightarrow [\sim ]{\rho _ f^{-1}} \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)}( f, f' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_ C, g \circ f')$

is given by the formation of right adjuncts (see Example 6.1.2.2 and Remark 6.1.2.4), and is therefore bijective by (Proposition 6.1.2.5). Assertion $(2)$ follows by a similar argument. $\square$

Variant 6.1.3.2. 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 the counit of an adjunction. Then:

$(1)$

For every $1$-morphism $f': C \rightarrow D$, the function

$\operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)}( f', f ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D) }( f' \circ g, \operatorname{id}_ D) \quad \quad \gamma \mapsto \epsilon (\gamma \circ \operatorname{id}_ g)$

is a bijection.

$(2)$

For every $1$-morphism $g': D \rightarrow C$, the function

$\operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)}( g', g ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D) }( f \circ g', \operatorname{id}_ D) \quad \quad \beta \mapsto \epsilon (\operatorname{id}_{f} \circ \beta )$

is a bijection.

Proof. Apply Proposition 6.1.3.1 to the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$. $\square$

Corollary 6.1.3.3. 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 $(\eta , \epsilon )$ be an adjunction between $f$ and $g$. Let $g': D \rightarrow C$ be another $1$-morphism of $\operatorname{\mathcal{C}}$. Then:

$(1)$

For every $2$-morphism $\eta ': \operatorname{id}_{C} \Rightarrow g' \circ f$, there is a unique $2$-morphism $\beta : g \Rightarrow g'$ for which $\eta '$ is equal to the composition $\operatorname{id}_{C} \xRightarrow {\eta } g \circ f \xRightarrow {\beta \circ \operatorname{id}_ f} g' \circ f$. Moreover, $\beta$ is an isomorphism if and only if $\eta '$ is the unit of an adjunction.

$(2)$

For every $2$-morphism $\epsilon ': f \circ g' \Rightarrow \operatorname{id}_ D$, there is a unique $2$-morphism $\gamma : g' \Rightarrow g$ for which $\epsilon '$ factors as a composition $f \circ g' \xRightarrow {\operatorname{id}_ f \circ \gamma } f \circ g \xRightarrow {\epsilon } \operatorname{id}_ D$. Moreover, $\gamma$ is an isomorphism if and only $\epsilon '$ is the counit of an adjunction.

Proof. We will prove $(1)$; the proof of $(2)$ similar. Let $\eta ': \operatorname{id}_{C} \Rightarrow g' \circ f$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. It follows from Proposition 6.1.3.1 that there is a unique $2$-morphism $\beta : g \Rightarrow g'$ satisfying $\eta ' = (\beta \circ \operatorname{id}_ f)\eta$. If $\beta$ is an isomorphism, then $\eta '$ is the unit of an adjunction by virtue of Remark 6.1.1.5. Conversely, suppose that $\eta '$ is the unit of an adjunction. To prove that $\beta$ is an isomorphism, it will suffice to show that for every $1$-morphism $g'': D \rightarrow C$, precomposition with $\beta$ induces a bijection $\operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g', g'' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g, g'')$. This is clear: we have a commutative diagram

$\xymatrix@R =50pt@C=30pt{ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g', g'' ) \ar [rr]^{ \beta } \ar [dr]_{\eta '} & & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g, g'') \ar [dl]^{\eta } \\ & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_{C}, g'' \circ f ), & }$

where the vertical maps are bijective by virtue of Proposition 6.1.3.1. $\square$

Proposition 6.1.3.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing $1$-morphisms $f,f': C \rightarrow D$ and $g,g': D \rightarrow C$. Let $(\eta , \epsilon )$ be an adjunction between $f$ and $g$, and let $(\eta ', \epsilon ')$ be an adjunction between $f'$ and $g'$. Then every $2$-morphism $\beta : f \Rightarrow f'$ determines a $2$-morphism $\beta ^{R}: g' \Rightarrow g$, which is uniquely determined by either of the following properties:

$(1)$

The diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{id}_{C} \ar@ {=>}[r]^-{\eta } \ar@ {=>}[d]^{\eta '} & g \circ f \ar@ {=>}[d]^{ \operatorname{id}_{g} \circ \beta } \\ g' \circ f' \ar@ {=>}[r]^-{ \beta ^{R} \circ \operatorname{id}_{f'} } & g \circ f' }$

commutes (in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$).

$(2)$

The diagram

$\xymatrix@R =50pt@C=50pt{ f \circ g' \ar@ {=>}[r]^-{\operatorname{id}_{f} \circ \beta ^{R} } \ar@ {=>}[d]^{\beta \circ \operatorname{id}_{g'} } & f \circ g \ar@ {=>}[d]^{ \epsilon } \\ f' \circ g' \ar@ {=>}[r]^-{\epsilon '} & \operatorname{id}_{D} }$

commutes (in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D)$).

Proof. It follows from Corollary 6.1.3.3 that there is a unique morphism $\beta ^{R}$ satisfying condition $(1)$. We will prove that $\beta ^{R}$ also satisfies condition $(2)$ (it is also uniquely determined by condition $(2)$, by virtue of Corollary 6.1.3.3). Note $(2)$ is equivalent to the assertion that $\epsilon '(\beta \circ \operatorname{id}_{g'})$ is the left adjunct of $\rho _{g}^{-1} \beta ^{R}$ with respect to $\epsilon$ (in the sense of Construction 6.1.2.1). By virtue of Proposition 6.1.2.5, this is equivalent to the assertion that $\rho _{g}^{-1} \beta ^{R}$ is the right adjunct of $\epsilon '(\beta \circ \operatorname{id}_{g'})$ with respect to $\eta$. This follows from the commutativity of the outer rectangle in the diagram

$\xymatrix@R =50pt@C=50pt{ g' \ar@ {=>}[r]^-{\lambda _{g'}^{-1}}_{\sim } \ar@ {=}[ddd] & \operatorname{id}_{C} \circ g' \ar@ {=>}[r]^-{\eta \circ \operatorname{id}_{g'} } \ar@ {=>}[d]_{\eta ' \circ \operatorname{id}_{g'} } & (g \circ f) \circ g' \ar@ {=>}[r]^-{ \alpha _{g,f,g'}^{-1}}_{\sim } \ar@ {=>}[d]_{ (\operatorname{id}_{g} \circ \beta ) \circ \operatorname{id}_{g'} } & g \circ (f \circ g') \ar@ {=>}[ddl]^{\operatorname{id}_{g} \circ (\beta \circ \operatorname{id}_{g'} )} \ar@ {=>}[dddd]^{ \operatorname{id}_{g} \circ \epsilon '(\beta \circ \operatorname{id}_{g'} ) } \\ & (g' \circ f') \circ g' \ar@ {=>}[r]^-{(\beta ^ R \circ \operatorname{id}_{f'}) \circ \operatorname{id}_{g'} } \ar@ {=>}[d]^{\sim }_{\alpha _{g',f',g'}^{-1}} & (g \circ f') \circ g' \ar@ {=>}[d]^{\sim }_{\alpha _{g,f',g'}^{-1}} & \\ & g' \circ (f' \circ g') \ar@ {=>}[r]^-{\beta ^{R} \circ (\operatorname{id}_{f'} \circ \operatorname{id}_{g'} ) } \ar@ {=>}[d]^{ \operatorname{id}_{g'} \circ \epsilon '} & g \circ (f' \circ g') \ar@ {=>}[ddr]^{\operatorname{id}_{g} \circ \epsilon '} & \\ g' \ar@ {=>}[r]^-{\rho _{g'}^{-1} } \ar@ {=>}[d]^{\beta ^{R}} & g' \circ \operatorname{id}_{D} \ar@ {=>}[drr]^{\beta ^{R} \circ \operatorname{id}_{\operatorname{id}_ D}} & & \\ g \ar@ {=>}[rrr]^{\rho _{g}^{-1} } & & & g \circ \operatorname{id}_{D} }$

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)$. Here the upper middle square commutes by virtue of condition $(1)$, the rectangle on the left commutes by virtue of the assumption that $(\eta ', \epsilon ')$ is an adjunction, and the commutativity of the rest of the diagram follows by the naturality properties of the associativity and unit constraints of the $2$-category $\operatorname{\mathcal{C}}$. $\square$

Notation 6.1.3.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of objects $C$ and $D$, and let $\underline{\operatorname{LHom}}_{\operatorname{\mathcal{C}}}(C,D)$ denote the full subcategory of $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)$ spanned by those $1$-morphisms $f: C \rightarrow D$ which admit a right adjoint $g: D \rightarrow C$. In this case, Corollary 6.1.3.3 guarantees that the $1$-morphism $g$ is determined uniquely up to isomorphism. We will sometimes abuse terminology by referring to $g$ as the right adjoint of $f$ and denoting it by $f^{R}$. The construction $f \mapsto f^{R}$ extends to a functor of categories $\underline{\operatorname{LHom}}_{\operatorname{\mathcal{C}}}(C,D)^{\operatorname{op}} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)$, which carries each $2$-morphism $\beta : f \Rightarrow f'$ to the $2$-morphism $\beta ^{R}: f'^{R} \Rightarrow f^{R}$ described in Proposition 6.1.3.4.

Warning 6.1.3.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ be a $1$-morphism of $\operatorname{\mathcal{C}}$. It follows from Corollary 6.1.3.3 that if $f$ admits a right adjoint $f^ R$, then $f^ R$ is characterized (up to canonical isomorphism) by the requirement that it represents the functor

$\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad g \mapsto \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D) }( f \circ g, \operatorname{id}_ D).$

Beware that it is possible for this functor to be representable by a $1$-morphism $g: D \rightarrow C$ which is not a right adjoint to $f$ (in which case $f$ cannot admit any right adjoint); see Warning 6.1.6.16.

The preceding discussion has an obvious counterpart for left adjoints:

Corollary 6.1.3.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $2$-morphisms of $\operatorname{\mathcal{C}}$, and let $(\eta , \epsilon )$ be an adjunction between $f$ and $g$. Let $f': C \rightarrow D$ be another $1$-morphism of $\operatorname{\mathcal{C}}$. Then:

$(1)$

For every $2$-morphism $\eta ': \operatorname{id}_{C} \Rightarrow g \circ f'$, there is a unique $2$-morphism $\beta : f \Rightarrow f'$ for which $\eta '$ is equal to the composition $\operatorname{id}_{C} \xRightarrow {\eta } g \circ f \xRightarrow {\operatorname{id}_ g \circ \beta } g \circ f'$. Moreover, $\beta$ is an isomorphism if and only if $\eta '$ is the unit of an adjunction.

$(2)$

For every $2$-morphism $\epsilon ': f' \circ g \Rightarrow \operatorname{id}_ D$, there is a unique $2$-morphism $\gamma : f' \Rightarrow f$ for which $\epsilon '$ factors as a composition $f' \circ g \xRightarrow {\gamma \circ \operatorname{id}_ g} f \circ g \xRightarrow {\epsilon } \operatorname{id}_ D$. Moreover, $\gamma$ is an isomorphism if and only $\epsilon '$ is the counit of an adjunction.

Proof. Apply Corollary 6.1.3.7 to the opposite $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. $\square$

Notation 6.1.3.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of objects $C$ and $D$, and let $\underline{\operatorname{RHom}}_{\operatorname{\mathcal{C}}}(D,C)$ denote the full subcategory of $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)$ spanned by those $1$-morphisms $g: D \rightarrow C$ which admit a left adjoint $f: C \rightarrow D$. In this case, Corollary 6.1.3.3 guarantees that the $1$-morphism $f$ is uniquely determined up to isomorphism. We will sometimes abuse terminology by referring to $f$ as the left adjoint of $g$ and denoting it by $g^{L}$. The construction $g \mapsto g^{L}$ determines an equivalence of categories $\underline{\operatorname{RHom}}_{\operatorname{\mathcal{C}}}(D,C) \rightarrow \underline{\operatorname{LHom}}_{\operatorname{\mathcal{C}}}(C,D)^{\operatorname{op}}$, which is homotopy inverse to the functor $f \mapsto f^{R}$ described in Notation 6.1.3.5.