Let $X$ be a topological space, $\mathcal{C}$ be a locally small category with "good properties" (such as having small inverse limit, small filtrant inductive limit...etc.) and $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$.


The + functor turns $\mathcal{F}$ to another presheaf $F^{+}$ such that: For any open subest $U$ of $X$, 


$$
F^{+}(U) = \varinjlim\_{\mathcal{U}} \ \mathcal{F}(\mathcal{U}),
$$
here $\mathcal{U}$ is a open convering of $U$ and $\mathcal{F}(\mathcal{U})$ is the equalizer of


$$
 \prod\_{V \in \mathcal{U}} \mathcal{F}(V) \rightrightarrows \prod\_{V\_1, V\_2 \in \mathcal{U}} \mathcal{F} (V\_1 \ \cap \ V\_2)
$$


One shows that $\mathcal{F}^{+}$ is a separated sheaf and if $\mathcal{F}$ is separated, $\mathcal{F}^{+}$ is a sheaf. So the sheafification is $\mathcal{F}^{++}$.


If $\mathcal{C}$ is a concrete category (everything are sets and maps), then the proof for these are clear.


My question is: For non-concrete categories, does one need some more conditions on $\mathcal{C}$ to prove these?


For example, in order to prove that $\mathcal{F}^{+}$ is separated, which is equivalent to


$$
\mathcal{F}^{+} (U) \rightarrow \mathcal{F}^{+} (\mathcal{U}) \mathrm{\ is \ a \ monomorphism},
$$


I found I need some conditions such as:


$$
\varinjlim\_{i} \ \mathrm{Hom}(A, X\_i) \rightarrow \mathrm{Hom}(A, Y) \mathrm{ \ is \ injective}
$$
$$ 
\varinjlim\_{i} \ \mathrm{Hom}(A, X\_i) \rightarrow \mathrm{Hom}(A, \varinjlim\_{i} \ X\_i) \mathrm{ \ is \ surjective}
$$


for some special filtrant inductive limits (those comes from open coverings of open subsets.)


The above conditioins are just from trying to mimic the proof when $\mathcal{C}$ is concrete. For example, for an element $x$ in a filtrant inductive limit, one can find some index $i$ and an element $x\_i \in X\_i$ such that $x$ is the image of $x\_i$.


I would like to know if such conditions are really necessary. If not, how one proves the properties of $\mathcal{F}^{+}$?




---


Edit: This question have been downvoted once, and I would like to know why. Is this question is not suitable for mathoverflow or it is just not the question of someone's favor?



+ functor (used to construct sheafification)'s property
You might be interested in the papers



> 
> P. Freyd, M. Kelly, *Categories of continuous functors I*, J. Pure Appl. Algebra 2 (1972), 1-18.
> 
> 
> J.W. Gray, *Sheaves with values in a category*, Topology 3 (1965), 1-18.
> 
> 
> 


