The classic way to encounter the theory of categories is via Set Theory via the typical definition we see for categories. We see all kinds of categories that are equivalent to the category of small categories. I wonder about presentations of the theory of categories. Is there any work done on cataloguing the various presentations. To facilitate an answer we may need to define a presentation of the theory. It may consist of a logical language or even the standard presentations of algebraic structures. For instance, a presentation of the theory of partial monoids would count as a presentation of Categories. The presentation should come with enough structure to analyze all small Categories.
Finally, one would expect a theory about all such presentations. Does this exist?