Let $\pi:X\to Y$ be a morphism of varieties. Let $V$ be a closed subvariety of $Y$, and denote by $\mathscr{H}^0_V$ the sections with support functor (see the excercises in section II.1 in Hartshorne's Algebraic Geometry for definition).
Is it true that $\mathscr{H}^0_V\circ \pi_{\ast}=\pi_{\ast}\circ \mathscr{H}^0_{\pi^{-1}(V)}$?
Let $U \subset Y$ be open and $\mathscr{F}$ a sheaf. The sections of the left side on $U$ are the sections of $\pi_*(\mathscr{F})(U) = \mathscr{F}(\pi^{-1}(U))$ which are non-zero in the stalks of $V \cap U$, but the stalks of $\pi_*(\mathscr{F})$ on $V \cap U$ are just the stalks of $\mathscr{F}$ on $\pi^{-1}(V \cap U)$. On the other hand, the sections on the right side are the sections of $\mathscr{F}(\pi^{-1}(U))$ which are non-zero in the stalks of $\pi^{-1}(U) \cap \pi^{-1}(V)$. So these are the same.