Example of isomorphic sheaves whose sections aren't isomorphic Are there examples of sheaves $\mathcal{F}$ and $\mathcal{G}$ such that $\mathcal{F} \cong \mathcal{G}$ but it is not necessarily true that $\mathcal{F}(U) \cong \mathcal{G}(U)$ for every open set $U$? I was sure there are, since a surjective morphism of sheaves doesn't always induce a surjective morphism of sections for every open set. However, I'm having a hard time coming up with examples.
If the answer turns out that no such examples exist, my next question would be whether or not a natural isomorphism of functors $F: A \rightarrow B$ and $G: A \rightarrow B$ would imply that $F(X) \cong G(X)$ for every object $X$?
 A: There are two possible definitions for natural isomorphism. For a natural transformation $\eta: F \to G$, both of the following statements make sense.

*

*Every arrow $\eta_X: F(X) \to G(X)$ is an isomorphism.

*There is a natural transformation $\theta: G \to F$ such that $\eta \theta$ is $Id_G$, the identity natural transformation on $G$, and $\theta \eta = Id_F$. In other words: $\eta$ is an isomorphism in the functor category where $F$ and $G$ belong to.

These two are equivalent. The direction $2 \implies 1$ is easy, for every $X$ we directly see that the inverse of $\eta_X$ is $\theta_X$. For the converse, $1 \implies 2$, there is only one sensible candidate for $\theta$. Namely we take $\theta_X$ to be $\eta_X^{-1}$, the inverse of $\eta_X$. We only need to check that $\theta$ is then indeed a natural transformation. So let $f: X \to Y$ be any arrow. Using naturality of $\eta$ we get
$$
F(f) \theta_X = \theta_Y \eta_Y F(f) \theta_X = \theta_Y G(f) \eta_X \theta_X = \theta_Y G(f).
$$
It might help to draw the relevant picture if the above is confusing.
So in particular, whenever two sheaves $F$ and $G$ are isomorphic through a natural isomorphism we get that each $F(U)$ and $G(U)$ are isomorphic for every $U$. You mention that an epimorphism of sheaves is not always componentwise surjective. An isomorphism is a very special kind of epimorphism, namely a split epimorphism. A split epimorphism of sheaves is always surjective on every component (nice exercise).
