Let $Y$ be a scheme and $\mathcal{G}$ a presheaf on any reasonble site over $X$; to have something concrete in mind let's take the small etale site $X_{\text{et}}$. Let furthermore $f:X \to Y$ any morphism of schemes.
There exist a presheaf pullback functor $f^P$ associating to $\mathcal{G}$ a presheaf on $X$ defined on sections by: For any etale $V \to X$ define
$$ f^P\mathcal{G}(V):= \varinjlim_U \mathcal{G}(U) $$
where the direct limit runs over etale $U \to Y$ through which the composition $V \to X \to Y$ factors. If $\mathcal{G}$ is a sheaf, the presheaf $f^P\mathcal{G}$ is in general not neccessarily a sheaf.
The "usual" pullback $f^*\mathcal{G}$ is defined as composition of $f^P$ with $s(-)$ the sheafification functor, a functor associating naturally a sheaf to a presheaf.
In general so far I know, the presheaf pullback functor not commutes with sheafification in the sense that for a presheaf $\mathcal{G}$
$$ s(f^P(s(\mathcal{G}))) \neq s(f^P(\mathcal{G})) $$
Or am I wrong with this claim and the two sheaves are always isomorphic? If not, can one in general at least say to how badly the sheaves $ s(f^P(s(\mathcal{G}))) $ and $s(f^P(\mathcal{G})) $ could differ from each over? Are there criteria when they coinside, beyond the "stupid" case when $f^P$ maps sheaves to sheaves, compare Milne's Etale Cohomology, Remark 2.14 (b), p 63 (the book, not the online script).