Are finite and finitely presented categories monadic over FinSet and Set?

Question

Finite categories, FC, are those consisting of finite sets of objects, morphisms and equations. Finitely presented categories, FPC, are those that can be generated from finitely many morphisms subject to finitely many equations. FPC form the category, CatFP, and FC forms FinCat, each with functors as morphisms. For a category, $C$, to be monadic over another category, $C2$, just means there is a monadic functor $C \rightarrow C2$. CatFP is not monadic over FinSet. So I am wondering if CatFP is monadic over Set, the category of sets and functions. Likewise, I am wondering if FinCat is monadic over FinSet, finite sets and functions. I am especially interested to know if either is Kleisli over their respective base. If so, what does the functor of the monad do?

I asked this over at Math OVERFLOW but was down voted. There is a comment that it may be true, so it may be worthwhile to try to figure this out.

Edit: It has been pointed out that I made a mistake in my definition of finitely presented categories. I am changing the question to include all the different options that I am interested in.