Multiplication of natural numbers can be understood as iterated addition, and we can understand binary Cartesian products as set-indexed coproducts; for sets $X$ and $Y$,


$$X\times Y\cong\coprod\_XY\cong\coprod\_YX.$$


The internalization of indexed coproducts is dependent sums, so we need these to ask the question



> 
> What categories with dependent sums does this relationship fail in for objects $X$ and $Y$? Is there a name for categories where this property holds?
> 
> 
> 



When aren't products iterated coproducts?
A discussion of the relation between dependent coproducts and cartesian products is found at


<https://ncatlab.org/nlab/show/dependent+sum>


