I think it's quite common pragmatic patterns have multiple names and parallel theoretical frameworks within to understand them. "A monad" is just a name to the

I think it's quite common pragmatic patterns have multiple names and parallel theoretical frameworks within to understand them. "A monad" is just a name to the applied pattern. There is nothing called "monad" in C++ for example, but you can write expressions in C++ that are isomorphic with monads. That does not mean that "monad" is the only correct name for the pattern, given people are familiar with different names and theoretical frameworks for the given pattern. The fact that the pattern is recognized as 'monad' is really usefull of course, since that implies all sorts of other things that may turn out to be usefull. But it does not mean IMO it's wrong to call it by some other popular name. The analogue in mathematics is rotation in a plane of the unit vector. We can present it as pair (cos(alpha),sin(alpha), a 2x2 matrix (m00,m01,m10,11), or even a complex number e^i alpha. "I'm rotating a vector" ... "no no, you are perfoming a matrix multiplication" ... "don't be silly, it's a complex number"...