Construction 8.6.4.20. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the category of elements of $\mathscr {F}$ (Definition 5.6.1.1); we identify objects of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ with pairs $(C,X)$, where $C$ is an object of $\operatorname{\mathcal{B}}$ and $X$ is an object of the category $\mathscr {F}(C)$. Let $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ denote the functor given on objects by $\mathscr {F}'(C) = \mathscr {F}(C)^{\operatorname{op}}$. We define a functor
as follows:
On objects, $\mathscr {K}$ is given by the formula $\mathscr {K}( (C,X'), (C,X) ) = \operatorname{Hom}_{ \mathscr {F}(C) }( X', X )$.
Let $f: (C,X) \rightarrow (D,Y)$ be a morphism in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and let $f': (C,X') \rightarrow (D,Y')$ be a morphism in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}'$ having the same image $u: C \rightarrow D$ in $\operatorname{\mathcal{B}}$. Let us identify $f$ and $f'$ with morphisms $g: \mathscr {F}(u)(X) \rightarrow Y$ and $g': Y' \rightarrow \mathscr {F}(u)(Y)$ in the category $\mathscr {F}(D)$. Then the function $\mathscr {K}(f',f): \mathscr {K}( (C,X'), (C,X) ) \rightarrow \mathscr {K}( (D,Y'), (D,Y) )$ is given by the composition
\[ \operatorname{Hom}_{ \mathscr {F}(C) }( X', X) \xrightarrow { \mathscr {F}(u) } \operatorname{Hom}_{ \mathscr {F}(D) }( \mathscr {F}(u)(X'), \mathscr {F}(u)(X) ) \xrightarrow { g \circ \bullet \circ g' } \operatorname{Hom}_{ \mathscr {F}(D) }( Y', Y ). \]