Lemma 89.24.2. Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\mathcal{C}_\Lambda $. Assume $U$ and $R$ are deformation functors. Then:

The quotient $[U/R]$ is a deformation category.

The tangent space of $[U/R]$ is

\[ T[U/R] = \mathop{\mathrm{Coker}}(ds-dt: TR \to TU). \]The space of infinitesimal automorphisms of $[U/R]$ is

\[ \text{Inf}([U/R]) = \mathop{\mathrm{Ker}}(ds \oplus dt : TR \to TU \oplus TU). \]

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