Problem: If $f \colon K \rightarrow L$ be a chain equivalence then $H_n (f) \colon H_n (K) \rightarrow H_n (L)$ is an isomorphism, in which, $H_n (K) = C_n (K) / B_n (K)$ is the homological module of $K$.
My question: Is there another way that is shorter to prove that? I just find out the simplest way that is, check that $H_n (f)$ be a homomorphism, injective, surjective, then conclude that it is an isomorphism.