For me when I was learning this the first problem was to grab what Mac Lane intended to say with "monoids" exactly (in monads are monoids such that blah) since

For me when I was learning this the first problem was to grab what Mac Lane intended to say with "monoids" exactly (in monads are monoids such that blah) since the default notion from algebra didn't match (unital semigroup). It turns out that he is speaking of monoid objects, which is a technical term as in here [1]. There it describes monoid object in a monoidal category, which is an unfortunate clash of two monoid-sounding things that need to be thought independently. One clarifies first what is the monoidal category, and only then asks what a monoid-object in that category is. In our case one starts with an arbitrary category C. The category of endofunctors has as objects functors F:C->C, and as morphisms natural tranformations mu:F1->F2. Endofunctors always compose, and that gives the endofunctor category a monoidal category structure: The monoidal product, or in other words the tensor product is the composition of endofunctors. One checks that this monoidal product verify the monoidal category axioms. So then one has established what monoidal category we talk aboout. It is in that monoidal category where one consider monoid objects, and then notices that they correspond to monads. [1] https://ncatlab.org/nlab/show/monoid+in+a+monoidal+category https://ncatlab.org/nlab/show/monoid+in+a+monoidal+category