The following result seems to be well known:
If T is a (V-)monad on a (V-)category C, then the forgetful functor $U^T \colon C^T \to C$ creates
- any limits that exist in C, and
- any colimits that exist in C and are preserved by T.
But I don't know of a published proof for general V. Toposes, Triples and Theories proves (1) for V = Set and leaves (2) for the special case of coequalizers as an exercise. Kelly's book doesn't mention monads at all. Lack, in Codescent objects and coherence, says that 'of course' (2) is true for V = Cat, but doesn't give a citation, while Blackwell, Kelly & Power, in 2-dimensional monad theory, say that (1) is 'well known' but don't give a reference either. And so on.
My questions are:
- Has a complete proof of this result (assuming it's true as stated) been published?
- Has its bicategorical analogue been treated?