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
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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