Let $U: Alg_T \to C$ be the forgetful functor from the category of algebras of $T: C \to C$ ($T$ could be a monad; I'm happy to think about the simpler case where $T$ is just an endofunctor or pointed endofunctor). Then a very simple diagram chase shows that

$U$ creates limits.

In other words, if I have a diagram in $Alg_T$ whose image in $C$ has a limit, there is an obvious way to lift the limit cone to $Alg_T$, and a straightforward diagram chase verifies that indeed, this is a limit in $Alg_T$.

This is all very easy, but I'm unhappy with the state of affairs for a few reasons:

I'm very lazy and I hate doing diagram chases.

This proof does not obviously generalize to other contexts. For one thing, I have to rehash variations on the same diagram chase for each of the cases where $T$ is a monad, an endofunctor, a pointed endofunctor, etc. For another, even if I stick with just monads, say, I have to rehash the same diagram chase if I want to generalize to other contexts such as enriched or internal category theory. Of course, doing the same work over and over is supposed to mean there's a bigger picture I'm missing.

I find it remarkable that in order for $U$ to create limits, one need not assume any kind of limit-preservation hypotheses about $T$. There's something to be explained, and the proof via diagram chase doesn't accomplish that.

The second point may have real weight -- I haven't checked very diligently, but it seems that it might not actually be known whether this this theorem remains true in full generality in the enriched context, for example! (Although a moment's reflection makes me think I could run the same diagram chase with no fuss if it weren't for point (1) above.) So here's my

**Question:** What is the conceptual reason for which monadic functors (and other forgetful functors from "categories of algebras") create limits?

categoryof the monad $\mathcal K(X,T)$ on the ordinary category $\mathcal K(X,Y)$. For reasonable examples of $\mathcal K$, limits in $Y$ are hopefully some kind of thing that turns into a limit in a category upon applying a representable functor, no? $\endgroup$ – Kevin Carlson Feb 17 at 2:47