Category of finite groups

The category of finite groups $\mathbf{FinGrp}$ is the full subcategory of the category of groups whose objects are finite groups. Its morphisms are all group homomorphisms between these objects; identities and composition are the corresponding functions.

Maps do not change when finiteness is recorded

For finite groups $G,H$ ,

\operatorname{Hom}_{\mathbf{FinGrp}}(G,H)=\operatorname{Hom}_{\mathbf{Grp}}(G,H).

Finiteness restricts the objects, rather than adding a condition to their group homomorphisms. The constant map to the target identity is included.