Let $\mathcal F, \mathcal G$ be two sheaves on the topological space X and $F,G$ be associated etale spaces, resp.
Assume that $\phi':F\to G$ is a continuous map (with respect to the etale space topology, of course), can we conclude that $\phi'$ induces a sheave map $\phi: \mathcal F \to \mathcal G$ ?
At first I thought this should be obvious. Let $U\subset X$ be open and $s\in \mathcal F(U)$. $\phi'(s_p)$ is defined for every $p\in U$. Then I can't see how to use the condition that $\phi'$ is continuous.
If this proposition is not correct, what condition can be added to make it correct?
You need to additionally assume that $\phi'$ is compatible with the projection maps $p:F\to X$ and $q:G\to X$, or in other words that $q\phi'=p$.
Given that, everything else is straightforward. If $s\in\mathcal{F}(U)$, that means $s:U\to F$ is continuous and $p(s(x))=x$ for all $x\in U$. Given such an $s$, we then define $\phi(s):U\to G$ as just the composition $\phi's$. This is indeed an element of $\mathcal{G}(U)$: it is continuous since $\phi'$ and $s$ are continuous, and $q(\phi'(s(x)))=p(s(x))=x$ for all $x\in U$.