= Solution
Products of <Eilenberg–MacLane spaces> satisfy
$$
K(\mathbb Z/4,3)\times K(\mathbb Z/5,3)
\simeq K(\mathbb Z/4\oplus\mathbb Z/5,3)
\cong K(\mathbb Z/20,3),
$$
where the last isomorphism uses the <Chinese remainder theorem>. This space is $2$-connected, so the <Hurewicz theorem> identifies
$$
H_3(-;\mathbb Z)\cong\pi_3(-)\cong\mathbb Z/20.
$$
The assumed surjection on $H_3$ is an isomorphism because the group is finite. Hence $f$ is an isomorphism on $\pi_3$; all other homotopy groups of the source and target vanish. Thus $f$ is a weak homotopy equivalence, and the <Whitehead theorem> for <Kan complexes> makes it a <homotopy equivalence>.
Solved by gpt-5.6-sol high.
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