The Stacks project

Lemma 8.4.3. Let $p : \mathcal{S} \to \mathcal{C}$ be a stack over the site $\mathcal{C}$. Let $\mathcal{S}'$ be a subcategory of $\mathcal{S}$. Assume

  1. if $\varphi : y \to x$ is a strongly cartesian morphism of $\mathcal{S}$ and $x$ is an object of $\mathcal{S}'$, then $y$ is isomorphic to an object of $\mathcal{S}'$,

  2. $\mathcal{S}'$ is a full subcategory of $\mathcal{S}$, and

  3. if $\{ f_ i : U_ i \to U\} $ is a covering of $\mathcal{C}$, and $x$ an object of $\mathcal{S}$ over $U$ such that $f_ i^*x$ is isomorphic to an object of $\mathcal{S}'$ for each $i$, then $x$ is isomorphic to an object of $\mathcal{S}'$.

Then $\mathcal{S}' \to \mathcal{C}$ is a stack.

Proof. Omitted. Hints: The first condition guarantees that $\mathcal{S}'$ is a fibred category. The second condition guarantees that the $\mathit{Isom}$-presheaves of $\mathcal{S}'$ are sheaves (as they are identical to their counter parts in $\mathcal{S}$). The third condition guarantees that the descent condition holds in $\mathcal{S}'$ as we can first descend in $\mathcal{S}$ and then (3) implies the resulting object is isomorphic to an object of $\mathcal{S}'$. $\square$

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