A morphism of set-valued functors $\eta: F \to G$ on $\mathcal{C}$ is called smooth if for all epimorphisms $B \to A$, the natural morphism $F(B) \to F(A) \times_{G(A)} G(B)$ is surjective.

Obviusly "smooth => formally" smooth for $\mathcal{C} = \mathrm{Sch}$.

Now my question: Does the converse hold?

My thoughts: 1. Assume the morphism is of finite presentation.

Assume it is in the local standard form: an étale morphism followed by an affine projection

It is clear that an affine projection is smooth in the above sense, so we have reduced the problem to étale morphisms.