A complex homomorphism $f:A_\ast\longrightarrow B_\ast$ is a family of maps $f_k:A_k\longrightarrow B_k$ such that $\partial^B_k\circ f_k=f_{k-1}\circ\partial^A_k$, that is, the diagram

is commutative

In this case, $f:A_\ast\longrightarrow B_\ast$ induces

linear maps, defined by $f_k[\alpha]=[f_k(\alpha)]$. The requirement $\partial^B_k\circ f_k=f_{k-1}\circ\partial^A_k$ ensures that the operation is well-defined

[Cohomological version]

A complex homomorphism $f:A^\ast\longrightarrow B^\ast$ is a family of maps $f^k:A^k\longrightarrow B^k$ such that $\mathrm{d}_B^k\circ f^k=f^{k+1}\circ\mathrm{d}_A^k$, that is, the diagram

is commutative

In this case, $f:A^\ast\longrightarrow B^\ast$ induces

linear maps, defined by $f^k[\alpha]=[f^k(\alpha)]$. The requirement $\mathrm{d}_B^k\circ f^k=f^{k+1}\circ\mathrm{d}_A^k$ ensures that the operation is well-defined