Let $f: S \to X$ be a surjective morphism between surfaces (so so $2$-dimensional, proper $k$-schemes) and $\mathcal{L} \in Pic(X)$ a globally generated invertible sheaf. Therefore for every $x \in X$ there exist a global section $s \in H^0(X, \mathcal{L})$ such that $s_x \neq 0$ as stalk in $\kappa(x)= \mathcal{L}_{X,x}/m_x$.
My question is how to show formally correctly that for surjective $f$ the pullback $f^*\mathcal{L}$ is also globally generated on $S$?
My thoughts: Since $f$ surjective for each $y \in S$ there exist $s \in H^0(X, \mathcal{L})$ with $s_{f(y)} \neq 0$. How to connect this information with the (locally) sheape of $f^*s$?
If $\alpha: E\to F$ is a map of (coherent) sheaves on $X$, it induces a map $f^*(\alpha): f^* E\to f^*F$ and if $\alpha$ is onto, so is $f^*(\alpha)$, since tensor product is right exact. In your case, $L$ is globally generated means, there exists a surjection $\mathcal{O}_X^n\to L$ and thus you get a surjection $\mathcal{O}_S^n\to f^*L$, which implies $f^*L$ is globally generated.