The definition of a monoid here is not the usual definition, is a new definition for the special case of a strict category as defined MacLane's Book Categories for the Working Mathematicean. Since you maybe thinking about composition of endofunctors and unit endofunctors you get a confused picture. The monad as a monoid in the category of endofunctors is a way to show that you can confuse people using two different definition of an usual concept and both give different results. I got this from (1), look for "main confusion": monoid in the category of endofunctors is defined in a new way, and is not the expected monoid in the set of all endofunctors.
The definition of monoid in monomial categories is on page 166 and Monads in a category are on page 133. As a math person I know what is a monoid (usual term) but I did not know what is a monoid in a monomial category (well I know now because is on page 166 of the book).
I wonder what's the point of using such a phrase, it doesn't help you to grok the concept of monad. It can help you to know that someone has given a new definition to sound cute, shame on them. By the way I admire MacLane as a math person, but people seem to use category theory to sell snake oil. Category theory is a tool to give names to some diagrams and properties that are used frequently to avoid repeating the same argument in proofs, is a dry (don't repeat yourself, as in the ruby motto). If you are an expert in category theory you can give short proofs of known facts. Category theory is like pipes in unix, you pipe diagrams to show properties. Grep, sed, awk analoges are functors, categories and natural transformations. The input is a collection of diagrams and the output is a new diagram that has a universal property and it receives a name and a collection of properties that are supposed to be useful to proof new theorems.
The phrase appears 6 chapters into a graduate mathematic text on category theory. If one reads the preceding chapters, it is a useful but pithy explanation for what a monad is, using terms which have all already been covered. Its use outside of that context is basically just a joke towards the Haskell community being overly mathematical.
The definition of monoid in monomial categories is on page 166 and Monads in a category are on page 133. As a math person I know what is a monoid (usual term) but I did not know what is a monoid in a monomial category (well I know now because is on page 166 of the book).
(1) https://stackoverflow.com/questions/3870088/a-monad-is-just-...