Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is cocomplete (Linton), and I think that we have the following **universal property**:

The category of cocontinuous functors $\mathsf{Mod}(T) \to \mathcal{D}$, where $\mathcal{D}$ is a cocomplete category, is equivalent to the category of cocontinuous functors $G : \mathcal{C} \to \mathcal{D}$ equipped with a right action $GT \to G$.

**Sketch of proof**: The free functor $F : \mathcal{C} \to \mathsf{Mod}(T)$ is cocontinuous and carries a right action $FT \to F$, induced by $\mu : T^2 \to T$. If $G : \mathcal{C} \to \mathcal{D}$ with $GT \to G$ is given, since every $T$-module $(X,a)$ has a canonical presentation $F(T(M)) \rightrightarrows F(M) \to (X,a)$ where the parallel arrows are given by the action $FT \to F$ and $F(a)$, and the right arrow is $a$, we have to define $\tilde{G} : \mathsf{Mod}(T) \to \mathcal{D}$ by $\tilde{G}(M,a):=$ coequalizer of $G(T(M)) \rightrightarrows G(M)$. Then $\tilde{G} F \cong G$ and $\tilde{G}$ preserves reflexive coequalizers (this is where I need that $T$ preserves reflexive coequalizers), so that $\tilde{G}$ is cocontinuous. QED

Actually I think there is also a version for cocomplete tensor categories and cocontinuous tensor functors; here $T$ should be a symmetric monoidal monad. This is the setting I'm actually interested in.

I don't really know much literature about category theory and in particular monad theory, but I suspect that this is known or even well-known. Is there any reference? (Again I need this in my dissertation and don't want to spam it with proofs of known facts.)

hascoproducts, but not that $\tilde{G}$ preserves them. $\endgroup$ – Mike Shulman Apr 19 '14 at 21:55