Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:

$$P:C^{op}to Set.$$

For every topology $J$ on $C$ we can generate a reflexive subcategory

$$Sh(C,J)subseteq Fun(C^{op},Set).$$

Part of the beauty/usefulness of this procedure is that the resulting objects are topoi and working in them is “easy”.

The first question is: why do we prefer $Set$ above all else? Why $Set$ seems to be the center of this construction?

Usually when consider functions/morphisms/functors targeting some structure, the set/category of functions/functors inherits this structure. (“Usually” our structure comes from products and limits so “usually” it works. I never worked with Hopf algebras to not use the usually.)

My guess is that it seems we want to “pullback” $Set$’s structure, that is a topos. Is this all there is to it?

The second question is why we do it for other categories as well? We consider sheaves of groups or rings. We will never get a topos, but we are nevertheless interested. And it seems that some people are interested even in sheaves over other less concrete categories. (I don’t have an explicit example.)

So the main question is: why do sheaves always seem to pop out? Why it seems that sheaves contain interesting information?

[EDIT]

It seems my question is ambiguous and noons gets it…

A group us not a sheaf, is a category. A ring is not a sheaf, is a category. A metric space is not a sheaf is a category. A posed is not a sheaf is a category ( in this case both enriched and not).

When we consider many concepts in math close to these concepts it seems that sheaves comes out pretty often. I do not know about stochastic processes, but I wouldn’t except them to be far away just because today none treats them as sheaves.

Most base theories are categories, because we need indexes. Most interesting construction on these theories are categories of sheaves. The question was why it is so.