“Rex” means “right exact”, which is the way abelian category theorists say “preserving finite colimits”. So cognoscenti in a rush say “rex functor” for “functor preserving finite colimits”, and use “Rex” for the category of categories with finite colimits and rex functors between these. Here I’m using it for the 2-category where we include natural transformations.
The analogous term “lex” is more commonly used, since “finite limits theories” are important in categorical algebra.

This leads to the riddle:

What’s the opposite of *Tyrannosaurus rex*?

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