Here we see the definition of an internal category in a monoidal category. We also know that endofunctor categories support a monoidal product which is actually functor composition. It is the case that a monoid in an endofunctor category is a monad, and so it makes sense that a comonoid is a comonad. We could take the route that an internal category here is a monad on the category of internal comonoids. It also looks to me like, if there is a comonad that is also a monad, we are done, we have an internal category. Does this make sense? Is it the case that if we have Frobenius monads (a monad that can be turned into a comonad) these are the internal categories?
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7$\begingroup$ I think the problem with such monoidal structures is that they generally do not preserve equalisers. $\endgroup$– Michal R. PrzybylekCommented Sep 1, 2014 at 22:28
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It seems that there was a bit of truth to this question. Here we see Spivak defining Categories as comonoids in the monoidal category $(Poly, \circ, y)$, section 2.5. As I point out, comonoids in an endofunctor category are comonads. I would guess that if you add a monad structure to your comonoids, you still have a category, but you have extra structure. It would be interesting to know what extra stuff you are endowing the category with if you add monad structure.