I'm reading some notes that compute the action of the boundary maps on the CW-complex $\mathbb{R}P^2\times\mathbb{R}P^2$. I know the chain maps for $RP^2$ itself are $d_0,d_1\equiv 0$, and $d_2=\cdot 2$.
Let the cell structures of the copies of the projective plane be $e_a^0\cup e_a^1\cup e_a^2$ and $e_b^0\cup e_b^1\cup e_b^2$. Then $(e_a^1,e_b^2)$ is a $3$-cell, and there is the computation
$$ d(e_a^1,e_b^2)=(0,e_b^2)-(e_a^1,2e_b^1)=-2(e_a^1,e_b^1) $$
where $d$ is the boundary map from the chain group of $3$-cells on the product complex.
There's no explanation for these rules, so I have to guess.
Is there some sort of product rule $$ d(U,V)=(d(U),V)+(-1)^{\deg U}(U,d(V)) $$ where $\deg U=n$ if $U$ is an $n$-cell? Also, do integer coefficients in either "coordinate" just move out in front?
Sorry for the vague question, but I can't find a reference for the algebraic rules of computing boundary maps of products of CW-complexes.