A concrete category over a category is a category together with a faithful functor . (The functor $U$ is sometimes called the forgetful functor or the underlying functor.)

A concrete category over $\Set$ is called a construct. (Here $\Set$ denotes the category of sets.)

This means that in a construct objects can be interpreted as sets and morphisms as maps.

Remarks:

1. An alternative meaning of a concrete category is that of a category with objects that have elements; such objects can be classes, semigroups, monoids, groups, groupoids, topological spaces, and so on.
2. Note also the Yoneda-Grothendieck Lemma that relates a category $\mathcal{C}$ to the functor category $\hat{\mathcal{C}}$ of contravariant functors from $\mathcal{C}$ to ${\bf Sets}$ , the category of sets.

Bibliography

1

J. Adámek, H. Herrlich, and G. Strecker.
Abstract and Concrete Categories.
Wiley, New York, 1990.