The Stacks project

Lemma 88.26.2. Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a deformation functor. Let $f: U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Then $U \times _{f, \mathcal{F}, f} U$ is a deformation functor with tangent space fitting into an exact sequence of $k$-vector spaces

\[ 0 \to \text{Inf}(\mathcal{F}) \to T(U \times _{f, \mathcal{F}, f} U) \to TU \oplus TU \]

Proof. Follows from Lemma 88.20.1 and the fact that $\text{Inf}(U) = (0)$. $\square$


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