In Katz & Mazur's book "Arithmetic Moduli of Elliptic Curves", available here, the definition of a full set of sections is given on page 33. I would like to make sure I understand the notations well.
The context is the following:
Let $S$ be a scheme and $Z/S$ a finite locally free scheme over $S$ of rank $N\geq 1$. This means that for any affine $S$-scheme $\operatorname{Spec}(R)\rightarrow S$, the $R$-scheme $Z_R/R$ obtained from $Z/S$ by base change is of the form $Spec(B)$ ($Z_R \rightarrow Spec(R)$ is finite hence affine), where $B$ is an $R$-algebra which as an $R$-module is locally free of rank $N$.
Any element $f\in B$ acts by multiplication to define an $R$-linear endomorphism of $B$. Because $B$ is a locally free $R$-module of rank $N$, we can speak of the characteristic polynomial of this endomorphism $$\det(T-f)=T^N-\operatorname{Trace}(f)T^{N-1}+\ldots+(-1)^N\operatorname{Norm}(f)$$
which is a monic polynomial in $R[T]$ of degree $N$.
(A justification for this last fact can be found on this related issue, for which any clarification is also welcome.)
Now, here comes the definition.
We say that a set of $N$ not necessarily distinct points $P_1,\ldots, P_N$ in $Z(S)$ is a full set of sections of $Z/S$ if for every affine $S$-scheme $\operatorname{Spec}(R)$, and for every function $f$ on $Z_R$ (ie $f \in B=H^0(Z_R,\mathcal{O})$), we have the equality of polynomials in $R[T]$ $$\det (T-f)=\prod _{i=1}^N(T-f(P_i))$$
All right, now my question may sound basic (and I apologize if it is), but I am unsure about how to make sense of $f(P_i)$. What does it mean? How can we apply an element of $B$ to a morphism $S\rightarrow Z$?
My guess is that such a morphism becomes $\operatorname{Spec}(R)\rightarrow Z_R$ after base change (it is defined by the composite morphism $\operatorname{Spec}(R)\rightarrow S \rightarrow Z$, together with the identity $\operatorname{Spec}(R) \rightarrow \operatorname{Spec}(R)$). Then, it induces a map on global section $B\rightarrow R$, which may then be applied to $f$ to get an element of $R$.
Is this construction what is really intended here? Moreover, if this is the case, wouldn't it make more sense to write $P_i(f)$ instead?
I thank you very much for you help.