These are not the monoids you're looking for...

When people first encounter the meme that monads are just monoids (in the category of endofunctors), some get the entirely wrong idea. Recall the monad laws:

• m >>= pure == m
• pure a >>= f == f a
• (m >>= f) >>= g == m >>= (\a -> f a >>= g)

If you squint a little, doesn’t this look a bit like a right and left identity law and an associativity law? Just like the monoid laws? Let’s squint a little harder and rewrite the laws in terms of Kleisli composition, that is, the following operator:

(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c)
(f >=> g) a = f a >>= g

The laws then become

• f >=> pure == f
• pure >=> g == g
• (f >=> g) >=> h == f >=> (g >=> h)

Very familiar. So, isn’t this the monoid we’re looking for? With pure being the identity and >=> the append operation?

NO!

These aren’t the laws of a monoid, but the laws of a category, the Kleisli category corresponding to the monad. A category is somewhat similar to a monoid – in fact, a monoid can be interpreted as a special case of a category. The difference is: In a monoid, any two elements can be appended, x <> y is always defined. In contrast, in a category, the composition of arrows¹ is defined only if the endpoint of the one arrow coincides with the starting point of the next, so the composition is a partial operation. You can see this from the type of (>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c). The composition f >=> g does not typecheck if the b in the return type of f is different from the b in the argument type of g.

You can read a full account about how a monad is a monoid in my article «A monad is just a monoid in the category of endofunctors, what’s the problem?».

The TL;DR is: The identity is pure, that much is correct, but the append operation is a function that is called join :: Monad m => m (m a) -> m a in Haskell. This seems incomprehensible at first, that’s because a monad is not in any way a monoid in the usual sense, but an arrow-theoretic generalization of a monoid, a thing called a monoid object, that can be defined in monoidal categories. If you want to know how this makes sense, seriously, read my article.