@Joemoeller Cartesian product is given by pairs of generic elements, thinking it should be given by pairs of global elements is just a misintuition

@boarders it sounds like you read the opposite of what I wrote.

@Joemoeller I re-read a couple of times, but no wiser as to what I have misinterpreted unfortunately

@boarders I may be misunderstanding your point, but I’m saying that because sometimes there are categories with a monoidal product given by pairs which is not the “limit product” as I’m suggesting to call it, the term “Cartesian product” can be conversationally confusing.

@Joemoeller maybe I’m just confused about what “given by pairs” means, unless it is just means that it is given by pairs of elements in some underlying set or something like that?

@Joemoeller my point was just that generic elements of the categorical product are given by pairs of generic elements and this fact means “Cartesian” is extremely appropriate (though I am not generally a fan of eponymous concepts, but neither do I see sense in top-down naming schemes since from where to do we look downwards from to take such a perspective)