Construction 5.2.6.1 (The Category of Elements). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor. 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 element of the set $\mathscr {F}(C)$.
If $(C,x)$ and $(C', x')$ are objects of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$, then a morphism from $(C,x)$ to $(C',x')$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is a morphism $f: C \rightarrow C'$ in the category $\operatorname{\mathcal{C}}$ for which the induced map $\mathscr {F}(f): \mathscr {F}(C) \rightarrow \mathscr {F}(C')$ carries $x$ to $x'$.
Composition of morphisms in $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is given by composition of morphisms in $\operatorname{\mathcal{C}}$.
We will refer to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the category of elements of the functor $\mathscr {F}$. Note that the construction $(C,x) \mapsto C$ determines a functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, which we will refer to as the forgetful functor.