$\require{AMScd}$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](https://www.sciencedirect.com/science/article/pii/002240497690013X), or [Flexible limits, p.4](https://core.ac.uk/download/pdf/82151786.pdf).


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"?
> 
> 
> 



Eilenberg-Moore category as a 2-dimensional limit
Yes, the Eilenberg–Moore object for a monad $T$ can be presented in terms of two [equifiers](https://ncatlab.org/nlab/show/equifier) 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](https://mathoverflow.net/a/352917) (see also [this question](https://mathoverflow.net/questions/413937/constructing-the-e-m-category-of-a-monad-out-of-inserters-and-equifiers)), for instance.


