Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

• Its 2-morphisms are the obvious thing.

**Question:** Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a

[insert adjective]functor from $C_1$ to $C_2$ ...

What is concrete equivalence?doi.org/10.1007/BF00878502, I think (I can't check right now) $\endgroup$ – David Roberts Aug 22 '17 at 9:53