"Functors between monads": what are these really called?

As far as I know, the first paper on this was:

Ross Street, The formal theory of monads. Journal of Pure and Applied Algebra 2 (1972), 149-168.

He called them monad functors. For the same thing but with the direction of the natural transformation reversed, he called them monad opfunctors. You can also consider the case where the natural transformation is an isomorphism, or even the identity, but I forget what he called them.

(If Street's wasn't the first paper on this, it was certainly early and very influential. And it's a beautiful paper.)

In a bid to try to make the terminology of 2-category theory more systematic, I called them lax maps of monads in my book Higher Operads, Higher Categories (CUP, 2004), and similarly colax etc. I don't know whether anyone else followed suit.

Recently I examined the thesis of one of Street's students, who I think used different terminology from what's in The formal theory of monads. I forget what it was, but I'll look it up if I remember.