Proposition. Let $T$ be a monad on $C$ and consider the forgetful functor $$ R^T \colon C^T \to C $$ from the category of Eilenberg-Moore algebras to $C$. This functor
- creates limits;
- creates colimits which are preserved by $T$ and $T^2$.
I have an example in mind of a functor which is created: direct limits (aka. colimits under directed diagrams) by the forgetful functor $$ U \colon \operatorname{Ring} \to \operatorname{Set}. $$
Question: are direct limits in Set preserved by
- the monad associated to the adjunction with Rings and its square,
- every monad and its square?
More generally, can I expect to prove that any direct limit (aka. colimit under directed diagam) is created by the forgetful functor from any Eilenberg-Moore algebra via the above proposition?