EDIT: It turns out I have read the statement wrongly. The statement in Wikipedia actually requires $\psi_{U_i}$ are isomorphisms of sheaves. With this, it's then clear how the result follows with my arguments below.
Original post:
I have just touched the elementary concept of sheaves. I found the following statement from Wikipedia:
A sheaf morphism $\psi:\mathcal{F}_1\rightarrow\mathcal{F}_2$ between two sheaves on topological space $X$ is a sheaf isomorphism iff there exists an open cover $\{U_i\}_{i\in I}$ of $X$ such that $\psi_{U_i}:\mathcal{F}_1(U_i)\rightarrow\mathcal{F}_2(U_i)$ are all isomorphisms
The only if direction is trivial, but how should I prove the if direction?
My intuition is that I should first argue the injectivity is guaranteed by the sheaf axiom (i) (locality axiom) and then prove the surjectivity by the gluing axiom of sheaf. But when I try to get into details to prove the injectivity, I'm stucked.
So suppose we have $\psi_{U_i}$ are all isomorphisms on an open cover $\{U_i\}_{i\in I}$, I want to first prove that for arbitrary open subset $V$ in $X$, $\psi_V$ is injective. To do so, I want to use the locality axiom. I take an open cover of $V$ by $\{V\cap U_i\}_{i\in I}$, and a section $f\in Ker(\psi_V)$, then since $\psi$ is a sheaf morphism, we have $$0=\psi_V(f)|_{V\cap U_i}=\psi_{V\cap U_i}(f|_{V\cap U_i})$$ So $f$ will be zero exactly by the locality axiom if I can prove that $\psi_{V\cap U_i}$ is injective, but unfortunately I have stucked at here for a while, I don't see why this is the case. Probably my argument is not the correct one, please give some helps, thank you!
Aside: I know nothing about category theory.