The Stacks project

Lemma 8.12.10. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$ satisfying the hypotheses and conclusions of Sites, Proposition 7.14.7. Let $p : \mathcal{S} \to \mathcal{C}$ and $q : \mathcal{T} \to \mathcal{D}$ be stacks. Then we have a canonical equivalence of categories

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Stacks}/\mathcal{C}}(\mathcal{S}, f_*\mathcal{T}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Stacks}/\mathcal{D}}(f^{-1}\mathcal{S}, \mathcal{T}) \]

of morphism categories.

Proof. For $i = 1, 2$ an $i$-morphism of stacks is the same thing as a $i$-morphism of fibred categories, see Definition 8.4.5. By Lemma 8.12.8 we have already

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Fib}/\mathcal{C}}(\mathcal{S}, u^ p\mathcal{T}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Fib}/\mathcal{D}}(u_ p\mathcal{S}, \mathcal{T}) \]

Hence the result follows from Lemma 8.8.3 as $u^ p\mathcal{T} = f_*\mathcal{T}$ and $f^{-1}\mathcal{S}$ is the stackification of $u_ p\mathcal{S}$. $\square$

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