TITLE: Reference request for Linton's theorems on equational theories
QUESTION [13 upvotes]: This is a reference request for the following "well-known" theorems in category theory:

There is an equivalence of categories between finitary monads on $\mathbf{Set}$ and finitary Lawvere theories (i.e. single-sorted finite product theories).

There is an equivalence of categories between monads on $\mathbf{Set}$ and infinitary Lawvere theories (i.e. singled-sorted product theories).

The category of models of a finitary/infinitary Lawvere theory is equivalent to the category of algebras of the corresponding monad.


I am not looking for proofs (I have written them up). I am looking for references which actually give proofs, so that I can cite them in a paper (instead of writing up the proof in the paper). Ideally, they should be classical references. I have scrolled through lots of papers which mention the theorems, and most of the time one (or several) of Linton's papers (let's give them letters) are cited:

E. Some aspects of equational theories, Proceedings of the Conference on Categorical
Algebra, Springer, 1966
F. An outline of functorial semantics, Seminar on triples and categorical homology theory,
Springer, 1969
A. Applied functorial semantics, Seminar on triples and categorical homology theory,
Springer, 1969

Sometimes, also the book "Toposes, Triples and Theories" by Barr & Wells is cited.
In E Linton only mentions 2) in the end of section 6, without proof. In fact, Linton proves a characterization theorem of the concrete categories of models of varietal theories (his name for infinitary Lawvere theories), via a version of the first isomorphism theorem, and he only mentions in passing that a combination with Beck's monadicity theorem yields the
equivalence to monads (but this detour is actually not necessary to get the equivalence; so this is not the simplest proof anyway).
I do not understand much what is going on in F (I find it very hard to read, also because of the typesetting), but it seems to deal with a much more general situation, and therefore I don't see where 1) or 2) is proven either. This is funny because both in the introduction of the Lecture Notes and in the introduction of A it is claimed that Linton proves 2) in F. Can perhaps someone help me to "decipher" F and explain where 2) is actually proved? The only result which looks similar is Lemma 10.2, but its proof is omitted... Maybe 3) follows from Theorem 9.3, but I don't see how.
The paper A focusses on monadicity criterions, and just mentions 2) in the introduction.
I could not find a proof in the book by Barr-Wells either. They talk about the history of these theorems in section 4.5 and attribute 1) and 2) to Linton's E and F.
I have found references with proofs of more general versions of 1), for example in the enriched case (Nishizawa, Power, Lawvere theories enriched over a general base), but it is probably awkward to cite such a paper for a classical result. I haven't found a published proof of 2) so far. The only thing which comes very close to 2) and 3) is the nlab article on algebraic theories: https://ncatlab.org/nlab/show/algebraic+theory, but the proof is sketchy.

REPLY [2 votes]: A detailed, self-contained and beginner-friendly proof of both theorems can now be found in my new paper on limit sketches in the appendix.