What's useful information available to us, when we state that : If $ f: X \to Y $ a morphism of schemes, and if $ \mathcal{F} $ denotes $ \mathcal{ O }_X $ - modules, and $ \mathcal { G} $ denotes $ \mathcal{ O }_Y $ - modules, then $ ( f^* , f_*) $ is a pair of adjoint functors, i.e that $ \mathrm{ Hom } ( f^* \mathcal{G }, \mathcal{F} ) \simeq \mathrm{ Hom } ( \mathcal{G} , f_* \mathcal{ O }_Y ) $ ?
What is meant concretly by saying that $ f^* $ and $ f_* $ are adjoints ?
Thank you in advance for you help.
$\newcommand{\sheaf}[1]{{\mathcal #1}}$ $\DeclareMathOperator{\Hom}{Hom}$
Aside from category-theoretic observations the isomorphism
$$\Hom(f^*\sheaf{G},\sheaf{F}) = \Hom(\sheaf{G},f_*\sheaf{F})$$
has the interpreation that both sides describe a set of morphisms
$$\psi_{U,V}: \sheaf{G}(V) \otimes_{\sheaf{O}_Y(V)} \sheaf{O}_X(U) \to \sheaf{F}(U)$$
for all $U \subseteq X$ and $V \subseteq Y$ open and $f(U) \subseteq V$, that are compatible in the sense that for $U' \subseteq U$ and $V' \subseteq V$ and $f(U') \subseteq V'$ we have
$$\rho^{\sheaf{F}}_{U,U'} \circ \psi_{U,V} = \psi_{U',V'} \circ \rho^{\sheaf{G}}_{V,V'} \otimes \rho^{\sheaf{O}_X}_{U,U'}$$
where $\rho^{\sheaf{F}}, \rho^{\sheaf{G}}$ are the restrictions in the respective sheafs.