I think the name “context wrapper” is meaningful representation. To me, (and i emphasise the to me part) It’s something that wraps on type to add more information to it.

For instance a time type could be wrapped in timezone Monad. To stop time being manipulated without the context (time zone) being corrupt, we use map to only allow specific changes to instance inside the monad.

I wrote a little blog post on my interpretation if anyone’s interested: https://blog.philliptaylor.net/?post=monads-for-oo-programmers.md

I guess all the good names were taken.

Categories are just the same as half of math, with less historical cargo so that they can all forced to look the same. But well, that historical stuff already took all the names.