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 $f: F\rightarrow G$ 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?