Let $\mathcal{F}$ be a presheaf valued in an arbitrary category $\mathcal{C}$ on a topological space $X$, with $\mathcal{C} $ has limits(or, $\mathcal{C}$ has equalizers so that a sheaf valued in $\mathcal{C}$ is well defined) and colimits(so stalks of $\mathcal{C} $ can be defined). What I am struggled with is how to construct the sheafification of $\mathcal{F} $ from its stalks $\mathcal{F}_x $. It can be shown that $\prod\mathcal{F}\colon U\rightarrow\prod_{x\in U}\mathcal{F}_x $ defines a sheaf from the presheaf $\mathcal{F} $, and the sheafification of $\mathcal{F} $ should somehow be the subsheaf of $\prod\mathcal{F} $, as in the case when $\mathcal{C}$ is the category of sets.
Now I am struggling with the construction of such subsheaf, and I am curious that is there any other condition on $\mathcal{C} $ such that sheafification of a sheaf can be defined. I would like to know is it possible to construct sheafification by the above procedure?
