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Definition (The Category of Elements: Covariant Version). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. 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 $u: \mathscr {F}(f)(X) \rightarrow Y$ is a morphism in the category $\mathscr {F}(D)$.

  • 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 $w: \mathscr {F}(g \circ f)(X) \rightarrow Z$ is the morphism of $\mathscr {F}(E)$ given by the composition

    \[ \mathscr {F}(g \circ f)(X) \xrightarrow { \mu _{g,f}^{-1}(X) } (\mathscr {F}(g) \circ \mathscr {F}(f))(X) \xrightarrow { \mathscr {F}(g)(u) } \mathscr {F}(g)(Y) \xrightarrow {v} Z, \]

    where $\mu _{g,f}: \mathscr {F}(g) \circ \mathscr {F}(f) \simeq \mathscr {F}(g \circ f)$ denotes the composition constraint for the functor $\mathscr {F}$.

We will refer to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the category of elements of $\mathscr {F}$.