Lemma 97.6.2. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $p$ and $q$ are formally smooth on objects, then so is the composition $q \circ p$.

Proof. This is formal. Let $U \subset U'$ be a first order thickening of affine schemes over $S$, let $z'$ be an object of $\mathcal{Z}$ over $U'$, let $x$ be an object of $\mathcal{X}$ over $U$, and let $\gamma : q(p(x)) \to z'|_ U$ be an isomorphism. As $q$ is formally smooth on objects there exist an object $y'$ of $\mathcal{Y}$ over $U'$, an isomorphism $\beta : y'|_ U \to p(x)$, and an isomorphism $\gamma ' : q(y') \to z'$ such that (97.6.0.1) is commutative. As $p$ is formally smooth on objects there exist an object $x'$ of $\mathcal{X}$ over $U'$, an isomorphism $\beta ' : x'|_ U \to x$, and an isomorphism $\gamma '' : p(x') \to y'$ such that (97.6.0.1) is commutative. The solution is to take $x'$ over $U'$ with isomorphism

$q(p(x')) \xrightarrow {q(\gamma '')} q(y') \xrightarrow {\gamma '} z'$

and isomorphism $\beta ' : x'|_ U \to x$. We omit the verification that (97.6.0.1) is commutative. $\square$

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