$S$ is a noetherian scheme, $\pi : X \rightarrow S$ is a relative projective morphism. $f: \mathcal{F} \rightarrow \mathcal{G}$ is a homomorphism of coherent sheaves over $X$, $\mathcal{G}$ is flat over $S$.
We can choose $n$ such that $H^1 (X_s,\mathcal{G}_s(n)) = 0 (\forall s\in S)$. $\phi : T \rightarrow S$ is an arbitrary morphism. We have the following commutative diagram.
$\begin{array}{ccc} X\times_S T & \stackrel{id_X \times_{S} \phi}{\longrightarrow} & X \\ {\scriptsize pr_2}\downarrow && \downarrow{\scriptsize \pi}\\ T &\stackrel{\phi}{\longrightarrow} & S\end{array}$
It seems that $\phi^* \pi_* \mathcal G(n) \cong {pr_2}_* (id_X \times_S \phi)^* \mathcal G(n)$, but I can't prove it. And I want to know why there is an induced homomorphism $\phi^* \pi_* \mathcal F(n) \rightarrow {pr_2}_* (id_X \times_S \phi)^* \mathcal F(n)$.