Definition 5.6.1.4 (The Category of Elements: Contravariant Version). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be a functor of $2$-categories (Definition 2.2.4.5). We define a category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ as follows:
The objects of $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ are pairs $(C, X)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\mathscr {F}(C)$.
Let $(C,X)$ and $(D,Y)$ be objects of $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$. Then 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 $u: X \rightarrow \mathscr {F}(f)(Y)$ is a morphism in the category $\mathscr {F}(C)$.
Let $(f,u): (C,X) \rightarrow (D,Y)$ and $(g,v): (D,Y) \rightarrow (E,Z)$ be morphisms in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$. Then the composition $(g,v) \circ (f,u)$ is the pair $(g \circ f, w)$, where $w: X \rightarrow \mathscr {F}( g \circ f)(Z)$ is the morphism of $\mathscr {F}(C)$ given by the composition
\[ X \xrightarrow { u } \mathscr {F}(f)(Y) \xrightarrow { \mathscr {F}(f)(v) } (\mathscr {F}(f) \circ \mathscr {F}(g))(Z) \xrightarrow { \mu _{f,g}(Z) } \mathscr {F}(g \circ f)(Z), \]where $\mu _{f,g}: \mathscr {F}(f) \circ \mathscr {F}(g) \simeq \mathscr {F}( g \circ f)$ denotes the composition constraint for the lax functor $\mathscr {F}$.
We will refer to $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ as the category of elements of the functor $\mathscr {F}$.