Suppose $(D_1,E_1;i_1,j_1,k_1)$ and $(D_2,E_2;i_2,j_2,k_2)$ are two exact couples, ie., there are exact sequences $D_1\xrightarrow{i_1}D_1\xrightarrow{j_1}E_1\xrightarrow{k_1}D_1$ and $D_2\xrightarrow{i_2}D_2\xrightarrow{j_2}E_2\xrightarrow{k_2}D_2$. It is known that spectral sequences could be deduced from exact couples. For simplicity, suppose the deduced spectral sequences converge after finite pages and the exact couples are over some field.
What is the tensor product of two exact couples so that the result behaves well for spectral sequences? I expect the result should be an exact couple like $(D,E_1\otimes E_2;i,j,k)$ for some $D,i,j,k$. But I don't know how to construct $D,i,j,k$ from $D_1,D_2,i_1,i_2,j_1,j_2,k_1,k_2$ suitably.
