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This answer states that the category of finitary monads is locally presentable and monadic over the category $\mathrm{Set}^{\mathbb{N}}$. Where can I find proof of this claim?

More generally: I've looked through the literature in nlab (finitary monad, monad, Lawvere theory) and found nothing about this category other than a proof of its equivalence with the category of Lawvere theories (written in several places). I would welcome links to any literature that discusses this category in more detail.

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    $\begingroup$ This may be going in a different direction than what you're looking for, but Durov's New Approach to Arakelov Geometry considers this category in Chapter 4, especially Sections 4.3 and 4.4. (I haven't read this in detail though.) $\endgroup$
    – Tobias Fritz
    Commented Apr 30, 2023 at 17:32

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These claims are proven more generally for the category $\mathrm{Mnd}_f(\mathscr A)$ of finitary monads on a locally presentable category $\mathscr A$ in Lack's On the monadicity of finitary monads. (This is the same Lack as in the linked answer.)

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    $\begingroup$ I was about to answer the same! $\endgroup$
    – fosco
    Commented Apr 30, 2023 at 17:38

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