My question is if homology commutes with taking tensor product. I believe in general it is not true but when the tensor product is with a projective module it is. I would like to take a look at a proof but I havent found one yet.

I was trying to proof the naturality of Kunneth exact sequence when I thought the second part of my question. If indeed homology commutes up to isomorphism, with taking the tensor product with a projective module, is this isomorphism natural?

This is: If $C'$ is a chain complex and $C_i$ is the chain complex which is $C_i$ in dimension, $i$ and zero in every other dimension and $C_i$ is a projective module, is this isomorphism natural?

$\;H_n ( C_i\bigotimes C')\cong C_i\otimes H_{n-i}( C').$