We know that, for any monad $T$, the Kleisli category $\mathcal{C}_T$ embeds into the Eilenberg-Moore category of $T$-algebras $\mathcal{C}^T$ as the full subcategory of free $T$-algebras. In the case of the monad for vector spaces, for example, this embedding is actually part of an equivalence of categories.
Are there nice categorical conditions on $T$, $\mathcal{C}_T$ or $\mathcal{C}^T$ that are sufficient to conclude that the full embedding $\mathcal{C}_T \hookrightarrow \mathcal{C}^T$ is essentially surjective, i.e that all $T$-algebras are essentially free?
I think Beck's monadicity theorem provides an answer, it gives sufficient conditions for a functor $G: D \to C$ to be 'monadic' (i.e. has a left adjoint $F$ such that $D$ is the space of $GF$-algebras).
Applying this theorem to the forgetful functor $U : C_T \to C$ of the Kleisli, quite a few conditions are trivially satisfied, resulting in the following necessary and sufficient condition for the embedding $C_T \hookrightarrow C^T$ to be essentially surjective
This condition is extremely technical, but if I understand the terms correctly a pair $f,g : A \to B$ has a split co-equaliser if there exists an arrow $h : B \to C$ such that the following commutes
$$ A \mathop{\rightrightarrows}^{f}_g B \mathop{\rightarrow}^h C $$ and there is a section $s$ of $h$ and a section $t$ of $f$ such that $g \circ t = s \circ h$.
Such a pair $f$, $g$ is called $U$-split if the above is true of $Uf$ and $Ug$.