What conditions on the coefficient category allow one to check isomorphisms for sheaves on stalks?

Let $mathscr{C}$ be a category which admits small limits and small filtered colimits.
For a topological space $X$,
one can define $operatorname{Sh}(X;mathscr{C})$ to be the subcategory of $operatorname{PSh}(X;mathscr{C}) = operatorname{Fun}(operatorname{Op}_X^{op},mathscr{C})$
generated by objects $F$ which satisfy the condition
$Gamma(U;F) cong lim Gamma(U_i;F)$ where ${U_i}$ is an open cover of U.

Now assume there is a morphism $f: Frightarrow G$ such that on each stalk $f_x$ is
an isomorphism.
What are the known conditions on $mathscr{C}$ which will imply $f$ an isomorphism in this situation?
In other words, what are the known conditions on $mathscr{C}$ that will make the functor
$I: operatorname{Sh}(X;mathscr{C}) rightarrow prod_{x in X} mathscr{C}$
which is given by $I(F) = (F_x)_{x in X}$ conservative?