Homomorphism of complexes

Definition
Given two chain complexes $$A$$ and $$B$$, a homomorphism of chain complexes $$h:A \to B$$ associates, for every $$n$$, a homomorphism $$h_n: A_n \to B_n$$ such that the $$h_n$$ commutes with the $$d$$ maps.

Homomorphisms at the level of cycle groups
A homomorphism of complexes induces a homomorphism at the level of their cycle groups. In other words, under the homomorphism from one chain group to another, the cycle group maps inside the cycle group of the other.

Homomorphism at the level of boundary groups
A homomorphism of complexes induces a homomorphism at the level of their boundary groups. In other words, under the homomorphism from one chain group to another, the boundary group maps inside the boundary group of the other.

Homomorphism at the level of homology groups
A homomorphism of complexes induces a homomorphism at the level of their homology groups. This follows from the above two facts: the boundary group maps to inside the boundary group, and the cycle group maps to inside the cycle group.