Joke explanation: "a comathematician is a device for turning cotheorems into ffee"

A category consists of objects and arrows between them (which can be associatively composed).

We can express various constructions and theorems with the help of categories, and when we switch the direction of all arrows, we arrive to the dual constructions and dual theorems, which are usually named by the prefix 'co', and which stay valid, as the proof/construction must go through with the reversed arrows the same way.

For example, the coproduct, i.e. the dual of the cartesian product of sets (in the category of sets and functions) is the disjoint union operation.

Of course, taking the dual is involutive, that is, 'co(co X)=X' whatever X is. That's why the dual of coffee is 'ffee', so the 'definition' of the comathematician is just the dual statement of the old joke.