I want to consider the sheaf valued in an arbitrary category (not only of sets, groups, modules and so on) on a topological space, using the language of étalé space.

In all references I am reading (Hartshorne's book, Harris' book, Vakil's note), only the sheaf of sets is considered in this way, although it is not hard to show that the same discussion works for sheaf of groups or modules. But how about the sheaf taking values in an arbitrary category?

For example, let $\mathcal{F}$ be a presheaf of sets on a topological space $X$. We can construct a topology on the disjoint union $F$ of $\mathcal{F}_p$ for all point $p \in X$ as well as a continuous map $\pi: F \rightarrow X$. Then the sheaf $\mathcal{F}'$ of sections of $\pi$ is the sheafification of $\mathcal{F}$.

But I cannot do the same discussion as above in the case where $\mathcal{F}$ is a presheaf valued in an arbitrary $\mathbf{C}$, because the construction of $\mathcal{F}'$ is based on topology but not on category.

That is, since $\mathcal{F}$ is a presheaf valued in the category $\mathbf{C}$, every $\mathcal{F}(U)$ and every $\mathcal{F}_p$ is an object of $\mathbf{C}$. Then if we assume that $\mathbf{C}$ has coproduct, the topological space $F =\coprod_{p \in X} \mathcal{F}_p$ can be also regarded an object of $\mathbf{C}$.

But $\mathcal{F}' (U)$ is defined to be the collection of sections of the map $\pi: F \rightarrow X$ on $U$, and none of categorical information is involved in this step. Then how can I claim that $\mathcal{F}'$ is a sheaf not only of sets but also valued in $\mathbf{C}$?

Of course, for the case of category of groups, rings, or $R$-modules, we can define addtions, productions, or $R$-actions of the sections, then I can show that $\mathcal{F}'$ is what I want. But what is the correct construction for an arbitrary $\mathbf{C}$?

Arbitrary$C$ is too general, since I can take, for instance $C$ could be (the coproduct completion of) a combinatorially-defined countable category whose objects have no geometric interpretation. – David Roberts 22 hours ago