The local nature of formal smoothness and an apparently extraneous hypothesis of finite presentation

In showing that formal smoothness is a local property, EGA seems to use the following presentation (and in fact, seems to contain a mistake according to this question). The proposition is as follows:

Proposition (EGA$ \ IV_{4}, 16.5.18)$ : Suppose $ X,Y$ are schemes over $ S$ and $ Y_0$ is a closed subscheme of $ Y$ defined by the ideal sheaf $ \mathscr I$ that is square $ 0$ . Assume also that $ Y$ is affine and $ \Omega^1_{X/S}$ is finitely presented.

We are given a map $ u_0: Y_0 \to X$ . Then, if $ U_\alpha$ is an open cover of $ X$ and we can lift $ u_0$ locally to maps $ u_\alpha$ defined on the appropriate open subsets of $ Y$ , we can find a global lift $ u: Y\to X$ .

My understanding of this proof is as follows. They show that the sheaf $ \mathscr P$ on $ Y$ defined by $ \Gamma(V,\mathscr P) = \{f: V \to X: f_0 = u_0: V/\mathscr I \to X\}$ is a torsor of the sheaf $ \mathscr G = \mathcal{Hom}_{Y_0}(u_0^{*}\Omega_{X/S}, \mathscr I) $ and since $ Y_0$ is affine, $ \mathscr P$ is the trivial torsor. Then, the $ 0$ section of $ \mathscr G$ gives the required lift.

Question: Assuming my outline of the argument above is correct, where does the finite presentation of $ \Omega_{X/S}$ get used?