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The Stacks project

Lemma 90.19.10. Let \varphi : \mathcal{F} \to \mathcal{G} be a morphism of categories cofibred in groupoids over \mathcal{C}_\Lambda satisfying (RS). Let x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k)). Then \varphi induces a k-linear map \text{Inf}_{x_0}(\mathcal{F}) \to \text{Inf}_{\varphi (x_0)}(\mathcal{G}).

Proof. It is clear that \varphi induces a morphism from \mathit{Aut}(x_0) \to \mathit{Aut}(\varphi (x_0)) which maps the identity to the identity. Hence this follows from the result for tangent spaces, see Lemma 90.12.4. \square

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