The Stacks project

Proof. This is formal. Let $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ be the directed limit of affine schemes $U_ i$ over $S$, let $z_ i$ be an object of $\mathcal{Z}$ over $U_ i$ for some $i$, let $w$ be an object of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U$, and let $\delta : p'(w) \to z_ i|_ U$ 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 $\delta : z \to z_ i|_ U$. Set $y_ i = q(z_ i)$ and $\gamma = q(\delta ) \circ \alpha : p(x) \to y_ i|_ U$. As $p$ is limit preserving on objects there exists an $i' \geq i$ and an object $x_{i'}$ of $\mathcal{X}$ over $U_{i'}$ as well as isomorphisms $\beta : x_{i'}|_ U \to x$ and $\gamma _{i'} : p(x_{i'}) \to y_ i|_{U_{i'}}$ such that (97.5.0.1) commutes. Then we consider the object $w_{i'} = (U_{i'}, x_{i'}, z_ i|_{U_{i'}}, \gamma _{i'})$ of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U_{i'}$ and define isomorphisms

and

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