As someone who studies both CS and pure math, it's quite interesting when I see posts that take a very operational reading of functors and monads, instead of its shorter and more precise categorical definition (as a mapping from one category to another that acts homomorphically over morphisms). That's not necessarily wrong, but it's an important thing to keep in mind when learning category theory through a programming context (e.g. Category Theory for Programmers by Milewski). When I tried to learn category theory in this way, past a certain point the abstractions felt too unnatural and arbitrary (e.g. lack of examples for adjunctions and free objects), whereas a good mathematical treatment of category theory would provide plenty of examples across different areas of mathematics such as algebra, topology and logic.

I strongly agree with valenterry's comment[1] here, that post gives is an approximate definition of functors, even omitting the functor law. In particular, it is explaining what an endofunctor is (in the context of FP, you mostly deal with one category anyway). Furthermore, even in Haskell, there is no such thing as the category of Haskell types[0] (because of things like divergence and partiality).

For the curious, I've recommended Program Design By Calculation[2] to a number of people to show how category theory can be used as a powerful tool to reason about programs in an FP context while not compromising on the mathematics (the target audience is undergrad CS majors).

EDIT: In particular, I have a bit of issue with this sentence after stating the signature of fmap, since it does not reference lawfulness.

> This is the full definition. There is no additional hidden machinery that Haskell is somehow automatically invoking when you "use a Functor".

[0] http://math.andrej.com/2016/08/06/hask-is-not-a-category/

[1] https://news.ycombinator.com/item?id=27638565

[2] https://www4.di.uminho.pt/~jno/ps/pdbc.pdf