Cocomplete concrete category, morphism of sheaves on a space is an isomorphism iff every induced map on stalks is an isomorphism

Question

Reading Hartshorne, I had this question:

Suppose $F$ and $G$ are $C$-valued sheaves on a space $X$, where $C$ is a cocomplete concrete category. Is it the case that $\phi:F→G$ is an isomorphism iff each induced map from $\phi$ on stalks is an isomorphism?

Hartshorne’s proof assumes $C(A,B)^x=C(A,B)\cap Set(A,B)^x$, where $C(A,B)^x$ denotes the isomorphisms from $A$ to $B$.

What if we don’t even assume $C$ is concrete?

Thank you!

Answer 1

In analogy with topos theory, we could say that the category $\text{Sh}(X;\mathcal{C})$ has enough points if your condition holds: For all $\mathcal{C}$-valued sheaves $F$ and $G$ on $X$, a map of sheaves $\varphi\colon F\to G$ is an isomorphism if and only if for all points $x\in X$, the induced map on stalks $\varphi_x\colon F_x\to G_x$ is an isomorphism.

The question of when a category of $\mathcal{C}$-valued sheaves on a space $X$ has enough points seems to be delicate. See Zhen Lin's answer here: A property of a sheaf in an arbitrary category, and see the nLab for information on the notion of a topos having enough points. Hopefully someone who knows more about this topic than I do can provide more information.