The Stacks project

Proof. Let us prove (2). An object of the category $\mathcal{S}'$ will be a pair $(U, \xi )$, where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $\xi$ is an isomorphism class of objects of $\mathcal{S}_ U$. A morphism $(U, \xi ) \to (V , \psi )$ is given by a morphism $x \to y$, where $x \in \xi$ and $y \in \psi$. Here we identify two morphisms $x \to y$ and $x' \to y'$ if they induce the same morphism $U \to V$, and if for some choices of isomorphisms $x \to x'$ in $\mathcal{S}_ U$ and $y \to y'$ in $\mathcal{S}_ V$ the compositions $x \to x' \to y'$ and $x \to y \to y'$ agree. By construction there are surjective maps on objects and morphisms from $\mathcal{S} \to \mathcal{S}'$. We define composition of morphisms in $\mathcal{S}'$ to be the unique law that turns $\mathcal{S} \to \mathcal{S}'$ into a functor. Some details omitted. $\square$