When are two morphisms of sheaves the same?

Question

Suppose I have two morphism $\phi_1, \phi_2 : F \rightarrow G$, where $F$ and $G$ are sheaves of sets on $X$. Is it enough to show that $\phi_1(X) = \phi_2(X)$ (i.e. as maps from $F(X)$ to $G(X)$) to show that the two morphisms between sheaves are the same? Thanks!

Answer 1

For example, pick any non-zero $F$ with no non-zero global sections, and look at the identity map $F\to F$ and at the zero map $F\to F$.

Answer 2

$F$ may have literally no global sections at all. For example, if $F$ is any connected non-trivial cover of the circle, then $F$ has no global sections but nonetheless has non-trivial automorphisms.

However, it is true that two morphisms of sheaves (on a topological space) are equal if and only if the induced morphisms of stalks are equal.