The laws for monoids in the category of endofunctors are:

And the Haskell monad laws are:

Left identity: return a >>= k = k a

Right identity: m >>= return = m

Associativity: m >>= (\x -> k x >>= h) = (m >>= k) >>= h

I'm assuming the latter is derived from the former, but how so? The diagrams basically say

join (join x) = join (fmap join x)
join (return x) = x
join (fmap return x) = x

How are these equivalent to the Haskell monad laws?

To show the >>=-monad laws from the join-monad laws, one needs to define x >>= y in terms of multiplication (join), unity (return), and functoriality (fmap), so we need to let, by definition,

(x >>= y) = join (fmap y x)

Left identity law

The left identity law then becomes

return a >>= k = k a

By definition of >>=, it is equivalent to

join (fmap k (return a)) = k a

Now, return is a natural transformation I -> T (where I is the identity functor), so fmap_T k . return = return . fmap_I k = return . k. We reduce the law to:

join (return (k a)) = k a

And this follows by the join laws.

Right identity law

The right identity law

m >>= return = m

reduces to, by definition of >>=:

join (fmap return m) = m

which is exactly one of the join laws.

I'll leave the associativity law for you to prove. It should follow using the same tools (join laws, naturality, functoriality).