I'm studying Algebraic Geometry from Gathmann's Notes and I don't find really satisfying how the product of abstract affine varieties is defined.
Page 34 (Gathmann's Notes) "[...] products involving X can be defined using any embedding of X in affine space (yielding a product that is unique up to isomorphisms)."
Does this mean that there is no canonical way of defining such product and I have to accept it as a definition up to isomorphism?