Two aspects of the presheaf theory of categories

The topos theory of categories, and the idea of presheaf representations still requires further clarification. I think we can split up the presheaf topos theory of categories into two parts:

• The topos theory of the category of categories $Cat$ which is now provided by the topos of compositional quivers. This topos is defined by chaining appropriate quivers in a composition manner.
• The topos theory of a general category $C$ which is provided by the Yoneda embedding $F: C \to Sets^{C^{op}}$ which fully and faithfully embeds any category into its topos of presheaves.

So basically, the theory of compositional quivers which I have defined is actually part of the topos theory of the category of categories $Cat$ while the topos theory of Yoneda's embedding of categories is part of the theory of individual and specific categories. The Yoneda embedding produces a different topos $Sets^{C^{op}}$ for every category.

The usefulness of the Yoneda's embedding doesn't mean that the idea of presheaf representations shouldn't be further explored. For one we could study the different Yoneda's embeddings of categories and how they relate to Grothendieck topoi and their topological properties, which I now has been done before. For another thing, its still useful to consider presheaf representations aside from the Yoneda's embedding for various reasons.

In particular, when considering something like the presheaf representation of algebraic structures, in our topos theory of universal algebra, it is best to consider presheaf topoi over finite index categories. As an algebraic structure is constructed out of a finite number of sets and functions, it should be defined as a presheaf over a category with a finite number of objects and morphisms.

Using the appropriate presheaf representation can lead to easier computations, and so that leaves the issue open. Whenever we consider an algebraic structure, we should immediately ask what kind of presheaf is it. The kind of presheaf it is, is determined how its sets and functions are combined. So that is why categories belong to the topos of compositional quivers, as that topos defines the composition law which is that morphisms $m: A \to B$ composed with morphisms $n: B \to C$ should produce morphisms of the form $n \circ m: A \to C$.

These different little details are going to be important in our implementations. One thing is for sure: everything is a presheaf and presheaves (respectively sheaves) are the most important objects in algebra and geometry. Algebraic structures are presheaves, like how categories are presheaves in the topos of compositional quivers. Geometric structures are sheaves such as schemes.

References:
Yoneda embedding