Definition. The category of elements [math-001C]

Let $A$ be a category, and $X \in \widehat{A}$. The category of elements of $X$ consists of

1. Each object is a pair $(a, s)$ such that $a \in A$ and $s \in X_a$.
2. Each morphism $(a,s) \xrightarrow{u} (b,t)$ is a morphism $u : a \to b$ in $A$, such that $u^*(t) = s$.

This category denotes $\int X$.