The Stacks project

Lemma 90.19.9. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Then $\text{Inf}_{x_0}(\mathcal{F})$ is equal as a set to $T_{\text{id}_{x_0}} \mathit{Aut}(x_0)$, and so has a natural $k$-vector space structure such that addition agrees with composition of automorphisms.

Proof. The equality of sets is as in the end of Remark 90.19.8 and the statement about the vector space structure follows from Lemma 90.19.7. $\square$

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