The Stacks project

Lemma 8.4.4. Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is a stack over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a stack over $\mathcal{C}$.

Proof. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$, $G : \mathcal{S}_2 \to \mathcal{S}_1$ be functors over $\mathcal{C}$, and let $i : F \circ G \to \text{id}_{\mathcal{S}_2}$, $j : G \circ F \to \text{id}_{\mathcal{S}_1}$ be isomorphisms of functors over $\mathcal{C}$. By Categories, Lemma 4.33.8 we see that $\mathcal{S}_1$ is fibred if and only if $\mathcal{S}_2$ is fibred over $\mathcal{C}$. Hence we may assume that both $\mathcal{S}_1$ and $\mathcal{S}_2$ are fibred. Moreover, the proof of Categories, Lemma 4.33.8 shows that $F$ and $G$ map strongly cartesian morphisms to strongly cartesian morphisms, i.e., $F$ and $G$ are $1$-morphisms of fibred categories over $\mathcal{C}$. This means that given $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $x, y \in \mathcal{S}_{1, U}$ then the presheaves

\[ \mathit{Mor}_{\mathcal{S}_1}(x, y), \mathit{Mor}_{\mathcal{S}_1}(F(x), F(y)) : (\mathcal{C}/U)^{opp} \longrightarrow \textit{Sets}. \]

are identified, see Lemma 8.2.3. Hence the first is a sheaf if and only if the second is a sheaf. Finally, we have to show that if every descent datum in $\mathcal{S}_1$ is effective, then so is every descent datum in $\mathcal{S}_2$. To do this, let $(X_ i, \varphi _{ii'})$ be a descent datum in $\mathcal{S}_2$ relative the covering $\{ U_ i \to U\} $ of the site $\mathcal{C}$. Then $(G(X_ i), G(\varphi _{ii'}))$ is a descent datum in $\mathcal{S}_1$ relative the covering $\{ U_ i \to U\} $. Let $X$ be an object of $\mathcal{S}_{1, U}$ such that the descent datum $(f_ i^*X, can)$ is isomorphic to $(G(X_ i), G(\varphi _{ii'}))$. Then $F(X)$ is an object of $\mathcal{S}_{2, U}$ such that the descent datum $(f_ i^*F(X), can)$ is isomorphic to $(F(G(X_ i)), F(G(\varphi _{ii'})))$ which in turn is isomorphic to the original descent datum $(X_ i, \varphi _{ii'})$ using $i$. $\square$

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