Presheaf representations of categories

I have recently described the representation of categories as composition quivers. This is useful for providing a topos from which we can consider and study categories. However, it is not strictly speaking the case that categories are equivalent to their composition quivers, or that things are implemented this way. There are two concepts:

• the category $C$ itself
• its presheaf representation

The two concepts don't need to coincide. While its reasonable to represent elementary categories without any additional structure as composition quivers, there are too many types of enriched categories for this to be the case in general. How for example does this apply to Ab-enriched categories. There are too many open questions for now.

Its reasonable for us to separate the two concepts for now. Instead, I suggest that we logicians study different presheaf representations so that we can apply categorical logic to different mathematical structures. In the same way that linear algebraists represent things with matrices to study them linear algebraically, we can represent mathematical structures with presheaves so that can be studied logically. This still needs to be researched further.