Eilenberg-Moore category as a 2-dimensional limit

Question

$\require{AMScd}$Given an endofunctor $F : C\to C$, its category of algebras is the inserter of $F$ and the identity functor. This means that there is a square $$\begin{CD} Alg(F) @>j>> C \\ @VjVV \Rightarrow@| \\ C @>>F> C \end{CD}$$ filled by a 2-cell $Fj\Rightarrow j$ and 1- and 2-terminal among all such.

Given a monad $T : C\to C$, one can build a similar diagram $$\begin{CD} EM(T) @>U>> C \\ @VUVV \Rightarrow@| \\ C @>>T> C \end{CD}$$ where $EM(T)$ is the Eilenberg-Moore category and $TU=UFU\Rightarrow U$ is $U\epsilon$, where $F\dashv U$ is the free-forgetful adjunction, and $\epsilon : FU\Rightarrow 1$ its counit.

However, this is not an inserter, but a subobject thereof: $EM(T)$ is smaller than the inserter of $T$ and $1$ because one has to require compatibility of endofunctor algebras with the monad structure.

Also, the usual presentation of $EM(T)$ as a limit is always described as more complicated: $EM(T)$ is the lax limit of the monad regarded as a lax functor from the point, and this is a 2-limit of the monad regarded as a simplicial object, as in Street, p.178, or Flexible limits, p.4.

Hence my question:

Is there any hope to present $EM(T)$ concisely, giving a universal property to the cell $U\epsilon$ above? Maybe an equalizer obtained from $Alg(T)$ (the endofunctor algebras of the underlying functor, i.e. $Ins(T,1)$)? Or an iterated inserter, or some other "simple" 2-limits combined together in a clever way that can be written down "finitely"?

Answer 1

Yes, the Eilenberg–Moore object for a monad $T$ can be presented in terms of two equifiers of the inserter $\mathbf{Ins}(T, 1)$. Denoting by $\phi \colon TU \Rightarrow U$, we equify $1_U$ and $\phi \cdot \eta U \colon U \Rightarrow U$, as well as $\phi \cdot T \phi$ and $\phi \cdot \mu U \colon T^2 U \Rightarrow U$. This serves to impose the two axioms of an algebra for a monad.

One explicit reference for this is §2.78 of Adámek–Rosický's Locally presentable and accessible categories. It is also described in this MathOverflow answer (see also this question), for instance.