Subcategory

Remarks

With the composition and identities of $\mathcal{C}$, a subcategory is a category. It is wide if $\mathrm{Ob}(\mathcal{D}) = \mathrm{Ob}(\mathcal{C})$, and full if $\Hom_{\mathcal{D}}(A, B) = \Hom_{\mathcal{C}}(A, B)$ for all its objects $A$, $B$. Example: inside the category of groups, the abelian groups with all homomorphisms between them form a subcategory. The identity of an abelian group is a homomorphism, and a composite of homomorphisms is a homomorphism. Non-example: in the category of sets, keep all sets but only the functions whose image has at most one point. Composites of such functions are again of this kind, so clause 2 holds, but the identity of $\set{0, 1}$ is missing, so clause 1 fails.