The Stacks project

Definition 8.2.2. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category, see Categories, Section 4.33. Given an object $U$ of $\mathcal{C}$ and objects $x$, $y$ of the fibre category, the presheaf of morphisms from $x$ to $y$ is the presheaf

\[ (f : V \to U) \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_ V}(f^*x, f^*y) \]

described above. It is denoted $\mathit{Mor}(x, y)$. The subpresheaf $\mathit{Isom}(x, y)$ whose values over $V$ is the set of isomorphisms $f^*x \to f^*y$ in the fibre category $\mathcal{S}_ V$ is called the presheaf of isomorphisms from $x$ to $y$.

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