Why is the ambiguity of the target category when first defining sheaves not a serious issue? I am taking notes on (pre)sheaves of topological spaces. This is my first time studying the subject. Some sources define them as functors to $Set$. Others as functors to $Ab$. Or sometimes $Mod_R$. Or $CRing$. They all clarify 'our category can be replaced by others (sets, abelian groups, ...)'
This kind of bothers me. In my notes, I wanted to avoid these ambiguities by just defining $C$-valued sheaves, for $C$ an arbitrary category. But this got annoying pretty quickly, since we soon need to worry about whether $C$ has certain properties (i.e. direct limits).
So I am going to concede and restrict myself to functors to a specific category, keeping in mind that that category 'can be replaced by others.' But I still want to say something about why this ambiguity is not a huge concern.
Unfortunately, I myself do not fully understand why this ambiguity basically gets brushed off. We can take $C=Set$ and this more-or-less allows us to formally analogize to other concrete categories. But then I get stuck with more questions, like 'what about non-concretizable categories?'
My questions:

*

*Is it simply the case that in practice, we end up working basically exclusively with 'nice enough' categories that we can not worry about the ambiguity of 'other categories' too much?


*Or, perhaps, is my impression that this ambiguity 'gets brushed off' actually incorrect, and at times it is necessary to go into specifics of what is/is not possible with (pre)sheaves taking values in a particular category?
I guess my fundamental problem is that I want a fairly uniform definition of a (pre)sheaf over a topological space. But I'm starting to think that I can't do this without bringing upon myself a lot of suffocating condition-checking, e.g. 'has direct limits.' (Suffocating, as in, it would be suffocating to restate lists of necessary conditions each time I define or describe something.)
 A: A few things to note. First, a sheaf on a space $X$ is just a functor $P \in Target^{\mathcal{O}_X^{op}}$ such that for all open covers $\bigcup\limits_{i \in } U_i = U$, the obvious diagram $P(U) \to \prod\limits_{i \in I} P(U_i) \rightrightarrows \prod\limits_{j \in I} \prod\limits_{k \in I} P(U_j \cap U_k)$ is an equalizer.
This means that the definition makes sense in any category $Target$ which has small products. This is all we need for the definition of a sheaf.
Edit: in the comments, Eric Wofsey pointed out that $Target$ having products isn't even necessary. We can instead consider the diagram formed by the individual $P(U_i)$ and $P(U_j \cap U_k)$ and assert that $P(U)$, together with restriction maps, is the limit of this diagram.
Things get rather interesting when we consider the special case $Target = Sets$. In this case, we have an adjunction $a \dashv i : Sh(X) \rightleftarrows Sets^{\mathcal{O}_X^{op}}$, with $a$ being the sheafification functor. Moreover, $a$ preserves finite limits.
We can use the fact that $a$ preserves finite limits to show that for any equational algebraic theory with finitary operations $T$, a sheaf over models of $T$ is just a $T$-object in the category of sheaves.
To be more precise, a sheaf of groups is just a group object in the category of sheaves, a sheaf of left $R$-modules is just an $R$-module object in the category of sheaves, a sheaf of monoids is just a monoid object in the category of sheaves, etc.
This means that to prove things about sheaves of (groups, rings, modules, etc.), we just need to prove things about sheaves of sets with a (group, ring, module, etc.)-structure. This dramatically simplifies things in many cases.
It is especially nice if you use the full power of the category of sheaves as a Grothendieck Topos. This means that you automatically get most constructions that can be performed in set theory for free. Most basic constructions (free groups, coequalizers of modules, kernels and images, etc) are constructions which can be performed in any topos trivially, so they carry over to $Sh(X)$ with no effort whatsoever. Those constructions which are less basic (eg constructions of infinitary coproducts) apply in any complete/cocomplete topos.
Edit: for the sheafification construction, we just go from $Models(T)^{\mathcal{O}_X^{op}} \to T$-objects in $Sets^{\mathcal{O}_X^{op}} \to T$-objects in $Sh(X)$ (using ordinary sheafification, which preserves finite limits and hence $T$-objects) $\to T$-valued sheaves on $X$.
A: Recall that there exists forgetful functors e.g from rings to sets, by post composing with this functor er induce a functor from sheaves of rings to presheafs of sets. Now the functor preserves limits (in the cases you mention) so the functor factors through sheafs of sets. Of course the forgetful functors are identities om objects and morphism, so se really do not have to be careful in these situations.
