Neither “Categorical product” nor “Cartesian product” are ideal descriptors imo. How are tensor products or coproducts not categorical? Cartesian is confusing when the product is not given by pairs, or when pairs gives a different monoidal structure.
I’m not sure why nobody afaict has used “limit product”.
@Joemoeller In defence of “categorical product” (even though it's not a term I really use myself), tensor products are *multi*categorical, and coproducts are not products.
I have said “limit product” in the past, when I needed to disambiguate.
@mudri coproducts do give a monoidal structure.
I’m not sure that historically that’s the distinction that “categorical” is making, but fair enough.
Joe Moeller
James Wood