Concrete category

https://en.wikipedia.org/wiki/Concrete_category

A concrete category is a category equipped with a faithful functor to 𝗦𝗲𝘁 category.

This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions.

Many important categories have obvious interpretations as concrete categories; for example, the category of topological spaces and the category of groups, and, trivially, also 𝗦𝗲𝘁 category itself. On the other hand, the homotopy category of topological spaces is non-concretizable i.e. it does not admit a faithful functor to the category of sets.

A concrete category, when defined without reference to the notion of a category, consists of a class of objects, each equipped with an underlying set; and for any two objects A and B a set of functions, called morphisms, from the underlying set of A to the underlying set of B. Furthermore, for every object A, the identity function on the underlying set of A must be a morphism from A to A, and the composition of a morphism from A to B followed by a morphism from B to C must be a morphism from A to C.

A relatively concrete category is a category that is equipped with a faithful functor to another category.