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 $C\_2$ ...
> 
> 
> 



Category of concrete categories
**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…*](https://link.springer.com/content/pdf/10.1007/BF00124121.pdf)) — 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](https://ncatlab.org/nlab/show/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. 


