The Stacks project

Lemma 89.20.4. Let $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ be maps of categories cofibred in groupoids over $\mathcal{C}_\Lambda $. If

  1. $\mathcal{F}$, $\mathcal{G}$ are deformation categories

  2. the map $T\mathcal{F} \to T\mathcal{G}$ is surjective, and

  3. $\mathcal{F} \to \mathcal{H}$ is smooth.

Then $\mathcal{F} \to \mathcal{G}$ is smooth.

Proof. Let $A' \to A$ be a small extension in $\mathcal{C}_\Lambda $ and let $x \in \mathcal{F}(A)$ with image $y \in \mathcal{G}(A)$. Assume there is a lift $y' \in \mathcal{G}(A')$. According to Lemma 89.20.3 all we have to do is check that $x$ has a lift too. Take the image $z' \in \mathcal{H}(A')$ of $y'$. Since $\mathcal{F} \to \mathcal{H}$ is smooth, there is an $x' \in \mathcal{F}(A')$ mapping to both $x \in \mathcal{F}(A)$ and $z' \in \mathcal{H}(A')$, see Definition 89.8.1. This finishes the proof. $\square$

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