I'm trying to typeset the naturality property of the cap product, I'm hoping to achieve something like this:
but the best I can do is something like this:
\begin{equation*}
\begin{tikzcd}
H_j(X,A\cup B; W) \arrow[d] &\otimes_\mathbb{Z} &H^i(X,A;V) \arrow[r]
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W) \arrow[d] \\
H_j(X,A\cup B; W) &\otimes_\mathbb{Z} &H^i(X,A;V) \arrow[u] \arrow[r]
& H_{j-i}(X,B;\overline{V} \otimes_\mathbb{Z} W)
\end{tikzcd}
\end{equation*}
I would like to group the two entries on either side of the tensor product at the beginning in a single entry, but then I wouldn't be able to get arrows to and from each one of them individually. I also don't want to change the general column spacing, just have those behave as a single entry but with arrows departing from each one. Thank you very much for any advice!