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 $z:U_1\Rightarrow U_2\circ F$.

• It's 2-morpihsms 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 $C_2$ ...