Let $X$ be a topological space and $\mathscr{F}, \mathscr{G}, \mathscr{H}$ be presheaves of Abelian groups on $X$. My question is that if $\phi :\mathscr{F} \rightarrow \mathscr{G}$, $\psi :\mathscr{G} \rightarrow \mathscr{H}$ are morphisms and $P \in X$, does $(\psi \circ \phi)_P = \psi_P \circ \phi_P$ hold?
My Approach :
Each element in $\mathscr{F}_P$ can be written by $s_P$ for some local section $s \in \mathscr{F}(U)$, where $U$ is an open neighbourhood of $P$. By definition, we have
$(\psi \circ \phi)_P(s_P) = [(\psi \circ \phi)(U)(s)]_P = [(\psi(U) \circ \phi(U))(s)]_P = \psi_P([\phi(U)(s)]_P) = \psi_P(\phi_P(s_P))$
I ask this question because I could find this formula in any (I known) textbook for algebraic geometry (is that too trivial? or just wrong?). Could someone tell me whether my arguments is true or not? Thank you very much!