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XXX is a pair $(C;V)$ or a category $C$ and a functor $V:\mathbf{Rel}\rightarrow C$.
How would you suggest to name XXX?
Probably I will add more requirements, such as $V$ being full or faithful, I’m yet unsure.
What’s about “category on top of $\mathbf{Rel}$”? Is this term taken?
category on top of Rel
no, this is a category under Rel. It’s an object in Rel/Cat. But more important is the concept of morphism between these things, as I can imagine several contenders.
And what’s about to name the functor $V$ “remembrance functor”? (It is mapping a less structured category into a more structured category, just reverse to “forgetful” functors, so “remembrance”.)
If not “remembrance” then how to call it?
“Category under Rel” sounds fine to me.
I second Mike’s opinion.
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