Let $C$ be a Category. Suppose that we are given collections

$Ob(D) \subseteq Ob(C)$, $\forall A,B \in Ob(D). D(A,B \subseteq C(A,B)$.

We say that $D$ is a subcategory of $C$ if

$A \in Ob(D) \Rightarrow id_A \in D(A,A)$,

$f \in D(A,B), g \in D(B,C) \Rightarrow g \circ f \in D(A,C)$

so that $D$ itself is a category.

In particular, $D$ is:

• A Full subcategory of $C$ if for any $A,B \in Ob(D), D(A,B) = C(A,B)$
• A Luff subcategory of $C$ if $Ob(D) = Ob(C)$.

For example, Grp is a full subcategory of Mon, and Set is a lluf subcategory of Rel.