Construction 8.6.1.9 (Conjugacy for Categories of Elements). Let $\operatorname{\mathcal{C}}$ be a category, let $\mathbf{Cat}$ denote the $2$-category of (small) categories, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. Let
denote the contravariant and covariant categories of elements of $\mathscr {F}$, respectively (see Definitions 5.6.1.4 and 5.6.1.1). Recall that objects of either category can be identified with pairs $(C,X)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\mathscr {F}(C)$. However, morphisms are defined differently:
A morphism from $(C,X)$ to $(D,Y)$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is a pair $(f,u)$ where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: \mathscr {F}(f)(X) \rightarrow Y$ is a morphism in the category $\mathscr {F}(D)$.
A morphism from $(C,X)$ to $(D,Y)$ in the category $\int ^{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F}$ is a pair $(g,v)$, where $g: D \rightarrow C$ is a morphism in the category $\operatorname{\mathcal{C}}$, and $v: X \rightarrow \mathscr {F}(g)(Y)$ is a morphism in the category $\mathscr {F}(C)$.
Let us identify the objects of the fiber product $( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ with pairs $(s: C' \rightarrow C, X)$, where $s: C' \rightarrow C$ is a morphism in $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\mathscr {F}(C')$. We define a functor We define a functor
as follows:
On objects, $T$ is given by the formula $T( s: C' \rightarrow C, X ) = (C, \mathscr {F}(s)(X) )$.
Let $(s: C' \rightarrow C, X)$ and $(t: D' \rightarrow D, Y)$ be objects of the category $( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. Unwinding the definitions, we see that a morphism from $(s: C' \rightarrow C, X)$ to $(t: D' \rightarrow D, Y)$ can be identified with triples $(f, f', u)$, where $f: C \rightarrow D$ and $f': D' \rightarrow C'$ are morphisms in $\operatorname{\mathcal{C}}$ satisfying $t = f \circ s \circ f'$, and $u: X \rightarrow \mathscr {F}(f')(Y)$ is a morphism in the category $\mathscr {F}(C')$. In this case, we define $T(f,f',u)$ to be the morphism $(f,v): (C, \mathscr {F}(s)(X) ) \rightarrow (D, \mathscr {F}(t)(Y) )$, where $v$ is the morphism in $\mathscr {F}(D)$ given by the composition
\begin{eqnarray*} (\mathscr {F}(f) \circ \mathscr {F}(s))(X) & \xrightarrow { (\mathscr {F}(f) \circ \mathscr {F}(s))(u) } & (\mathscr {F}(f) \circ \mathscr {F}(s) \circ \mathscr {F}(f'))(Y) \\ & \simeq & \mathscr {F}( f \circ s \circ f')(Y) \\ & = & \mathscr {F}(t)(Y), \end{eqnarray*}where the unlabeled isomorphism is supplied by the composition constraints for the functor $\mathscr {F}$.