# Kerodon

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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

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

is commutative, 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 $d: T \rightarrow D$.