I am trying to prove (a special case of) Theorem 3.6.3 which is Kunneth formula for complexes stated in Weibel on page 88:
Let $C_\bullet$ and $D_\bullet$ be chain complexes of abelian groups and assume that $C_\bullet$ is projective. Then, the following s.e.s splits: $$0 \to \bigoplus_{p+q = n} H_p(C_\bullet) \otimes H_q(D_\bullet) \to H_n(C_\bullet \otimes D_\bullet) \to \bigoplus_{p+q=n-1}Tor_1(H_p(C_\bullet),H_q(D_\bullet)) \to 0.$$
I don't know how to approach the proof. Any hints and/or references will be appreciated.
Edit: A proof is given in Hatcher Theorem 3B.5. but is for when both $C_\bullet$ and $D_\bullet$ are free (here same as projective), not just $C_\bullet$ (which is what we are assuming here). For this more restrict assumption, Hatcher refers the reader to "A Course in Homological Algebra" by Hilton and Stammbach. I checked that book and there indeed is a proof. I am wondering if there are other (shorter) proofs, or proofs that would start with both complexes being free and extend it to just one of them being free.