The Stacks project

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 $w$ be an object of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U$, and let $\delta : p'(w) \to z'|_ U$ be an isomorphism. We may write $w = (U, x, z, \alpha )$ for some object $x$ of $\mathcal{X}$ over $U$ and object $z$ of $\mathcal{Z}$ over $U$ and isomorphism $\alpha : p(x) \to q(z)$. Note that $p'(w) = z$ hence $\delta : z \to z|_ U$. Set $y' = q(z')$ and $\gamma = q(\delta ) \circ \alpha : p(x) \to y'|_ U$. As $p$ is formally smooth on objects there exists an object $x'$ of $\mathcal{X}$ over $U'$ as well as isomorphisms $\beta : x'|_ U \to x$ and $\gamma ' : p(x') \to y'$ such that (97.6.0.1) commutes. Then we consider the object $w = (U', x', z', \gamma ')$ of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U'$ and define isomorphisms

and

These combine to give a solution to the problem. $\square$