TITLE: Category of concrete categories
QUESTION [9 upvotes]: 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 $C_2$ ...

REPLY [10 votes]: Concrete functor is established in the literature for the related notion where the natural transformation is an isomorhpism (see e.g. Porst 1996 Concrete Categories Are Concretely Equivalent if…) — i.e. the sub-2-category of the slice 2-category of CAT over Set on faithful functors.
Your 2-category is similarly the sub-2-category on faithful functors of the colax slice 2-category of CAT over Set.  So it seems very natural to call your notion colax concrete functors, though as far as I can find this term hasn’t been used before.  A lax concrete functor would be the same thing but with the transformation in the other direction.