Typo: In “note that $M\to\mathbf{M}(N)$ is a functor” I think it should be $M\mapsto\mathbf{M}(M)$ instead. This was also pointed out in [#7169](https://stacks.math.columbia.edu/tag/05AB#comment-7169).

In the proof, after “pick any ordinal $\alpha$ whose cofinality is greater than $|U|$,” I think we later use that $\alpha$ is actually a limit ordinal when we say “then we see that $\varphi$ factors through $M_{\alpha'}(M)$ for some $\alpha'<\alpha$ by Proposition $\ref{079F}$.” Maybe one should mention some of this? ($\alpha$ will always be a limit ordinal since $|U|\geq 1$ and by Sets, Comment [#9498](https://stacks.math.columbia.edu/tag/05N3#comment-9498)),

Also, in the transfinite recursion, we are not only defining $\mathbf{M}_\alpha(M)$ for each $\alpha$ but also a natural transformation $\mu_{\alpha,\beta}:\mathbf{M}_\alpha\to \mathbf{M}_\beta$ for each $\alpha<\beta$, right? And such that $\mu_{\alpha,\beta,M}$ is injective for all $M\in\operatorname{ob}\mathcal{A}$ and $\mu_{\beta,\gamma}\circ\mu_{\alpha,\beta}=\mu_{\alpha,\gamma}$ for $\alpha<\beta<\gamma$. In other words, we are getting a functor $\operatorname{Ord}\to\operatorname{Fun}_\mathrm{inj}(\mathcal{A},\mathcal{A})$, where $\operatorname{Ord}$ is the totally ordered class of ordinal numbers and $\operatorname{Fun}_\mathrm{inj}(\mathcal{A},\mathcal{A})$ is the wide subcategory of $\operatorname{Fun}(\mathcal{A},\mathcal{A})$ where the morphisms are the natural transformations whose components are all injective maps. After thinking for a while, the following is what I came up with.

For each ordinal number $\alpha$, denote $S(\alpha)$ to the set of ordinal numbers $\leq\alpha$ (this is $S(\alpha)=\alpha\cup\{\alpha\}$, the successor of $\alpha$). For each ordinal number $\alpha$, we want to define a functor
$$\begin{aligned} \mathbf{M}^{\leq\alpha}:S(\alpha) &\to\operatorname{Fun}_\mathrm{inj}(\mathcal{A},\mathcal{A})\\ \beta&\mapsto{}\mathbf{M}^{\leq\alpha}_\beta\\ \gamma\leq\beta &\mapsto\mu^{\leq\alpha}_{\gamma,\beta} \end{aligned}$$
such that $\mathbf{M}^{\leq\alpha}|_{S(\beta)}=\mathbf{M}^{\leq\beta}$ for all $\beta\leq\alpha$. The desired functor $\operatorname{Ord}\to\operatorname{Fun}_\mathrm{inj}(\mathcal{A},\mathcal{A})$ will be obtained by taking the union of the functors $\mathbf{M}^{\leq\alpha}$.

We do so by transfinite recursion in $\alpha$.

- **Base step.** For $\alpha=0$, define $\mathbf{M}^{\leq 0}:S(0)\to\operatorname{Fun}_\mathrm{inj}(\mathcal{A},\mathcal{A})$ as $\mathbf{M}^{\leq 0}_0=\operatorname{id}_\mathcal{A}$.

- **$\mathbf{+1}$ case.** Suppose such a functor $\mathbf{M}^{\leq\alpha}$ is defined. For an ordinal $\alpha$, define $\mathbf{M}^{\leq\alpha+1}:S(\alpha+1)\to\operatorname{Fun}_\mathrm{inj}(\mathcal{A},\mathcal{A})$ as
$$\begin{align*} \mathbf{M}^{\leq\alpha+1}|_{S(\alpha)} &=\mathbf{M}^{\leq\alpha}\\ \mathbf{M}^{\leq\alpha+1}_{\alpha+1} &=\mathbf{M}\circ\mathbf{M}^{\leq\alpha+1}_\alpha\\ \mu^{\leq\alpha+1}_{\alpha,\alpha+1} &=\mu_{\mathbf{M}^{\leq\alpha}_\alpha}\\ \mu^{\leq\alpha+1}_{\beta,\alpha+1} &=\mu^{\leq\alpha+1}_{\alpha,\alpha+1} \circ \mu^{\leq\alpha+1}_{\beta,\alpha}&\text{for }\beta<\alpha, \end{align*}$$
where $\mu:\operatorname{id}_\mathcal{A}\to\mathbf{M}$ is the natural transformation whose component at $M$ is the morphism $M\to\mathbf{M}(M)$ in the cocartesian square that defines $\mathbf{M}(M)$, and in the second-to-last equality we are using the whiskering notation from Categories, Section $\ref{003D}$. Since $\mu$ is injective, so is the morphism in the second-to-last equality; hence also the morphism in the last equality is injective.

- **Limit case.** Suppose $\alpha\neq 0$ is a limit ordinal and that $\mathbf{M}^{\leq\beta}$ is defined for all $\beta<\alpha$. We define $\mathbf{M}^{\leq\alpha}$, on the one hand, by setting $\mathbf{M}^{\leq\alpha}|_{\alpha}$ to be the union of the functors $\mathbf{M}^{\leq\beta}$ for $\beta<\alpha$, and on other hand, by
$$\tag{1} \mathbf{M}^{\leq\alpha}_\alpha=\underset{\beta<\alpha}{\operatorname{colim}}\mathbf{M}_\beta^{\leq\alpha}.$$
For $\beta<\alpha$, the map $\mu_{\beta,\alpha}^{\leq\alpha}$ is defined to be the leg at $\beta$ of the limiting cocone $(1)$. Since $\mathcal{A}$ is AB5, by the Lemma in Comment [#9497](https://stacks.math.columbia.edu/tag/079F#comment-9497), $\mu_{\beta,\alpha}^{\leq\alpha}$ is injective.