If you are switching around arrows, wouldn't it be more natural to have box topology as the categorical dual of disjoint union topology since both require the image or preimage of an open set under canonical maps to be open.
I do understand that box topology is not a product since the unique map produced is not necessarily continuous.
What does it mean to be categorical dual to something?
"What does it mean to be categorical dual to something?"
If you have some dual notions like products and coproducts or more generally limits and colimits then duality pretty much means reversing all arrows and keeping defining properties. That also shows that products are turned into coproducts in the opposite category and vice versa.
The product topology has the property that the product map $f \colon X \rightarrow \prod Y_i$ is continuous iff all maps into the factors $f_i \colon X \rightarrow Y_i$ are continuous which fails for the box topology in general. This is dual to the coproduct map $ f \colon \coprod Y_i \rightarrow X$ being continuous iff all the $f_i \colon Y_i \rightarrow X$ are continuous.
The box topology can still be useful to create counterexamples though.