Let $f:X\rightarrow Y$ be a morphism of schemes. Let $f^*$ be the pullback functor from the sheaves on the big etale site (or small etale/Zariski site depending on the context!) of $Y$ to $X$. Let $L$ be the forgetful functor from the sheaves on the big etale site to the small Zariski site. When is it true that $L$ and $f^*$ commute? (For what morphisms $f$? is it true if $f$ is the inclusion of the generic point?)
What happens if the big etale site changes to the small etale site?