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?

Handbook of Categorical AlgebraVolume 2, 4.3 (Borceux). I would be surprised if the proof didn't work for general $V$... $\endgroup$ – Dylan Wilson Jul 22 '13 at 21:43