My question is about the proof of 8.4/2 in "Neron models." The claim is that if $f\colon X \rightarrow S$ is a proper flat morphism of finite presentation such that $H^2(X_s, \mathscr{O}_{X_s}) = 0$ for every $s \in S$, then the relative Picard functor $\mathrm{Pic}_{X/S}$ is formally smooth.

The proof proceeds by taking an affine test scheme $Z$ and a closed subscheme $Z_0 \subset Z$ defined by an ideal of square zero and proving that the map $$R^1(f\times_S Z)_* \mathscr{O}^*_{X\times_S Z} \rightarrow R^1(f \times_S Z_0)_* \mathscr{O}^*_{X\times_S Z_0}$$ is a surjection of sheaves (on the etale or the fppf site). My question is: how is this enough for the claimed formal smoothness? It seems to me that what one wants is the surjectivity of $$\mathrm{Pic}_{X/S}(Z) = H^0(Z, R^1(f\times_S Z)_* \mathscr{O}^*_{X\times_S Z}) \rightarrow H^0(Z_0, R^1(f \times_S Z_0)_* \mathscr{O}^*_{X\times_S Z_0}) = \mathrm{Pic}_{X/S}(Z_0),$$ which doesn't seem to follow formally because a surjection of sheaves need not be a surjection on global sections.