The category $\mathbf{Cat}$ agrees with the category of small categories and functors. See it as a $1$-category. It is possible to show for instance that it is bicomplete but not regular (why it is not regular? Where can I find a detailed proof?) Actually, it is quite difficult to find a full list of properties satisfied by $\mathbf{Cat}$. Is there some book or paper where such a category has been fully investigated? I'm interested for instance in extremality, regularity, effectiveness, strongness of monics and epics, stability of epics under pullbacks, effectiveness of equivalence relations and so on. If possible, can you provide a full list of properties of the $1$-category $\mathbf{Cat}$ (with reference)?
Moreover, it is possible to show that it is equivalent to the category $\mathbf{Mon}$ of monoids. Furthermore, in his book, Spivak showed that such a category is equivalent to the category of Schemas (it is a definition taken from database theory) Are there other categories that have been proved to be equivalent (or isomorphic) to $\mathbf{Cat}$?