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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