The Stacks project

Lemma 98.21.1. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let

\[ \xymatrix{ B' \ar[r] & B \\ A' \ar[u] \ar[r] & A \ar[u] } \]

be a commutative diagram of $S$-algebras. Let $x$ be an object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$, let $y$ be an object of $\mathcal{Y}$ over $\mathop{\mathrm{Spec}}(B)$, and let $\phi : f(x)|_{\mathop{\mathrm{Spec}}(B)} \to y$ be a morphism of $\mathcal{Y}$ over $\mathop{\mathrm{Spec}}(B)$. Then there is a canonical functor

\[ \textit{Lift}(x, A') \longrightarrow \textit{Lift}(y, B') \]

of categories of lifts induced by $f$ and $\phi $. The construction is compatible with compositions of $1$-morphisms of categories fibred in groupoids in an obvious manner.

Proof. This lemma proves itself. $\square$

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