[This answer](https://mathoverflow.net/a/55356/148161) states that the category of finitary monads is locally representable 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](https://ncatlab.org/nlab/show/finitary+monad), [monad](https://ncatlab.org/nlab/show/monad#References), [Lawvere theory](https://ncatlab.org/nlab/show/Lawvere+theory#references)) 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.