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.

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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 $\sigma_{i+1}:SX_{i+1}\to X_{i+1}$ 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 $(S,\eta)$.

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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.

> 1. Is there something subtly (or not so subtly) wrong with this observation?

> 2. 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?

> 3. 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?


<cite authors="Richard Garner" mrnumber="2506256" cite="_Appl. Categ. Structures_ **17** (2009), no. 3, 247--285">_Richard Garner_, MR 2506256 [**Understanding the small object argument**](http://dx.doi.org/10.1007/s10485-008-9137-4), _Appl. Categ. Structures_ **17** (2009), no. 3, 247--285.</cite> 

<cite authors="G. M. Kelly" mrnumber="589937" cite="_Bull. Austral. Math. Soc._ **22** (1980), no. 1, 1--83">_G. M. Kelly_, MR 589937 [**A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on**](http://dx.doi.org/10.1017/S0004972700006353), _Bull. Austral. Math. Soc._ **22** (1980), no. 1, 1--83.</cite>