This answer 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, 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.