The Stacks project

Lemma 96.5.1. Let $S$ be a scheme. Let

\[ \xymatrix{ \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Y}' \ar[r]^ g & \mathcal{Y} } \]

be a $2$-cartesian diagram of categories fibred in groupoids over $S$. Then we have a canonical isomorphism

\[ g^{-1}f_*\mathcal{F} \longrightarrow f'_*(g')^{-1}\mathcal{F} \]

functorial in the presheaf $\mathcal{F}$ on $\mathcal{X}$.

Proof. Given an object $y'$ of $\mathcal{Y}'$ over $V$ there is an equivalence

\[ (\mathit{Sch}/V)_{fppf} \times _{g(y'), \mathcal{Y}} \mathcal{X} = (\mathit{Sch}/V)_{fppf} \times _{y', \mathcal{Y}'} (\mathcal{Y}' \times _\mathcal {Y} \mathcal{X}) \]

Hence by (96.5.0.1) a bijection $g^{-1}f_*\mathcal{F}(y') \to f'_*(g')^{-1}\mathcal{F}(y')$. We omit the verification that this is compatible with restriction mappings. $\square$

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