If Monad is just a Monoid Object in the Category of Endofunctors; where a Monoid is just a construct(product/pair) of a Set and a binary operator, having identity, closure, and associativity;
and a Comonad is just a Comonoid Object in the Category of Endofunctors; eg a Monad with all arrows reversed.
According to this diagram,
Does this mean that there also exist Magmad, a Magma Object in the category of Endofunctors; a Semigroupad, a Semigroup Object in the category of Endofunctors; and a Groupad, a Group Object in the category of Endofunctors; a Comagmad, a Cosemigroupad, and a Cogroupad as well?
I'm curious because I tried googling these terms but the only instances of Groupad seem to be mispellings of Grouped, so I am curious if it actually makes sense or not, and I'm curious why all the hype about Monads in functional programming without any respect for Magmads, Semigroupads, and Groupads.
If there is a Monad which also has inverse, does it make it a Groupad? And would any data structure that has a binary operation that creates an element of the same data structure be reasonable to call a Magmad? And if it isn't quite a Monad since it lacks identity, would it be reasonable to call it a Semigroupad?