Looking at how pub fn cartesian_product<I, J>(mut i: I, j: J) -> Product<I, J> initialises Product, one can see that it yields the first item in i.
Wouldn't it be better if Product.a_cur was an Option<Option<I::Item>> and initialized as None, rather than initializing it by starting the consumption of i?

This may cause problems if iteration over i has side effects.

Same with Product.b too. No reason to clone it ahead of time.

I haven't looked deep into at the implementation of multi_cartesian_product but it may be a similar situation there too.