Free monad sequence versus colimit over injections of ordered sets Kelly describes a constructive procedure for building the algebraically free monad on a pointed endofunctor. Garner gives a concise summary, which I partially review here for convenience.

Let $V$ be a cocomplete category (feel free to make other strong assumptions about $V$ if I have omitted them) and $(S,\eta)$ a pointed endofunctor of $V$. Define $X_0=\mathrm{id}_V$, $X_1=S$, and $\sigma_0:SX_0\to X_1$ as the identity. Then assuming $\sigma_0,\ldots,\sigma_i$ and $X_0,\ldots,X_{i+1}$ are defined, define $X_{i+2}$ and $\sigma_{i+1}:SX_{i+1}\to X_{i+2}$ as the coequalizer of the following diagram:
$\require{AMScd}$
\begin{CD}
SX_i @>\sigma_{i}>>X_{i+1}\\
@VS\eta X_iVV @V\eta X_{i+1}VV\\
SSX_i@>S\sigma_i>>SX_{i+1}
\end{CD}
Then putting $\{X_i\}$ together via the maps $X_{i}\xrightarrow{\eta X_i} SX_{i}\xrightarrow{\sigma_i} X_{i+1}$, the resulting diagram
$$
X_0\to X_1\to \cdots
$$
is the (finite part of the) free monad sequence for $(S,\eta)$. If we make what seems to be a fairly strong convergence assumption, then the colimit $X_\omega$ of the sequence is the (algebraically) free monad on $(S,\eta)$.

On the other hand, another thing one could thing to do is to consider $(S,\eta)$ as generating a semicosimplicial functor $S_*$ where $S_n= S^n$ and the coface maps are given by $S^i\eta S^j$. Then one could consider the colimit of the diagram made up by coface maps up to a certain point:
$$
S_0\to S_1\rightrightarrows S_2\to\cdots \to S_n
$$
Call this colimit $Y_n$.
Inductively it looks to me like $X_i\cong Y_i$ and so $X_\omega\cong Y_\omega$.
Here are my questions.

  
*
  
*Is there something subtly (or not so subtly) wrong with this observation?
  
*The presentation of $Y_n$ and $Y_\omega$ seems "more symmetric." Assuming the answer to question 1. is "no," is there a good reason to prefer the presentation of $X_n$ and $X_\omega$? Is this because things will go badly as soon as one needs to extend to infinite ordinals in the free monad sequence?
  
*If the answer to question 1. is "no," is there a reference that uses the $Y_n$ presentation to build this sequence where I can see in detail whether there are secret assumptions I am suppressing?

Richard Garner, MR 2506256 Understanding the small object argument, Appl. Categ. Structures 17 (2009), no. 3, 247--285. 
G. M. Kelly, MR 589937 A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bull. Austral. Math. Soc. 22 (1980), no. 1, 1--83.
 A: Let's talk about free monoid constructions in a monoidal category $(\mathcal{V}, \otimes, I)$. Free monad constructions are a special case, when we take $\mathcal{V}$ to be a monoidal category of endofunctors. There are at least two flavors: the free monoid on an object $X$, and the free monoid on a pointed object $I \to X$.
It's pretty easy to construct the free monoid functor $T: \mathcal{V} \to \mathcal{V}$ in typical monoidal categories -- just take $T: X \mapsto \sum_{n=0}^\infty X^{\otimes n}$, or in the pointed case, $T: I \to X \mapsto \varinjlim_{n\to \infty} X^{\otimes n}$ using the map $I \to X$ to define semicosimplicial connecting maps. The issue is with defining multiplication. We want to say that $TX \otimes TX = \varinjlim_m X^{\otimes m} \otimes \varinjlim_n X^{\otimes n} = \varinjlim_{m,n}X^{\otimes m+n}$, and then define multiplication $\mu_X: T^2 X \to TX$ by the universal property of the colimit, setting $X^{m+n} \to T^2 X \overset{\mu_X}{\to} TX$ equal to $X^{m+n} \to TX$, where the unlabeled maps are the canonical colimit inclusions. If you think about this, you'll see it's the usual formula for multiplication in a free monoid. The diagrams all commute, so we can define this map $\mu_X$ with no problem.
The problem is that the domain of $\mu_X$ is $\varinjlim_{m,n} X^{\otimes m+n}$, and in order to identify this with $T^2 X$, we had to commute a colimit past $\otimes$ -- we needed the functor $TX \otimes(-): \mathcal{V} \to \mathcal{V}$ to commute with the colimit defining $TX$. This is fine if $\otimes$ is (right) closed, in which case $TX \otimes(-)$ commutes with all colimits, so these simple constructions work in the familiar cases of free monoids in categories like $\mathsf{Set},\mathsf{Ab}$, etc.
But right closedness is simply not true in a category of endofunctors $\mathcal{V} = [\mathcal{C},\mathcal{C}]$. The condition becomes slightly different in this setting -- if you start with $F$ and define $M(C) = \varinjlim_n F^n C$, you want to argue that $M^2(C) = \varinjlim_m F^m (\varinjlim_n F^n(C)) = \varinjlim_{m,n} F^{m+n}(C)$, for which it suffices for $F$ to commute with the colimits defining $M$. Unfortunately, this is not often the case, although I believe the semisimplex category is at least sifted, so if $F$ commutes with sifted colimits you might be in luck.
Otherwise I don't know a shortcut to reading Kelly. I think the trick to reading his paper on a first pass must be to keep in mind that you can take the factorization system he uses to be $(M,E)$ = (all maps, isos), and then "preserving $E$-tightness of a cocone" just means preserving a colimit. What I currently understand is that Kelly gives free constructions endofunctor -> pointed endofunctor -> well-pointed endofunctor -> monad. I'm currently most mystified by the middle one.
