Definition(from P276 of Introduction to Homological Algebra by Rotman):
Let $S = \{E, p, X\}$ and S' = {E', p', X} be etale-sheaves over a space $X$. An etale map f: S $\to$ S' is a continuous map f: E $\to$ E' such that p'f = p and each f|E$_x$ is a homomorphism. Then $\text{Hom}_e$$_t$ is an additive abelian group if we define f + g: E $\to$ E' by (f + g)(e) = f(e) + g(e).
I am really new to sheaves. How is this a group. I couldn't prove if it is closed under addition: p'(f+g)(e) = p'(f(e)+g(e)) and then how to proceed as p' here is only a continuous map between top spaces? Additionally, how to see that f+g is continuous when it is defined that way? Any help would be greatly appreciated!