Set $f:Y \to X$ be a morphism between schemes and $s \in \Gamma(X, \mathcal{O}_X)$ be a global section.
The map $f$ induces a functor $f^*$ (so called pullback functor) that pulls back $\mathcal{O}_X$-modules over $X$ to $\mathcal{O}_Y$ modules.
Futhermore for the global section $s$ I often encounter the notation $f^*s$ ("pullback of a global section")
Using the well know adjunction correspondence between pullback and pushforward we get for arbitrary sheaf $\mathscr F$ a natural morphism of sheaves $\mathscr F\to f_* f^*\mathscr F$. Obviously this induces in functorial sense a map between global sections
$$H^0(X,\mathscr F)\to H^0(X,f_* f^*\mathscr F)=H^0(Y,f^*\mathscr F)$$
which exactly maps $s$ to $f^*s$.
Formally that's ok. My problem is how it concretely looks like.
The most easiest case that $X= Spec(R), Y= Spec(A)$ and $f$ is induced by the ring map (=map between global sections) $\varphi_f: R \to A$.
Therefore every $r \in R$ is a global section.
How concretely in this case the pullback $f^*r \in A$ looks like and how to derive/conclude it?