Concrete category

Remarks

Example: the category of groups with the functor sending a group to its underlying set is concrete. Two homomorphisms $G \to H$ with the same underlying function are equal, so $U$ is faithful. Non-example: the pair (groups, $P$), where $P$ sends every group to a one-point set, is not a concrete category. The homomorphisms $x \mapsto 0$ and $x \mapsto x$ of $\Z$ are different but have the same image under $P$, so $P$ is not faithful. Every small category $\mathcal{C}$ is concretizable. Send $A$ to the set of all morphisms with codomain $A$, and $f \colon A \to B$ to the map $g \mapsto f \circ g$. This functor is faithful, since $f$ is the image of $1_A$.