The Categories $\mathrm{Set}\times\mathrm{Set}$ and $\mathrm{Set}+\mathrm{Set}$

Question

I'm thinking about the following problem. In $\mathrm{Cat}$ I can form the product $\mathrm{Set}\times\mathrm{Set}$. Elements are tuples, say $(A,X)$. I think that inner products and coproducts are just formed pairwise, i.e. $(A,X)\times(B,Y)=(A\times B,X\times Y)$ and $(A,X)+(B,Y)=(A+B,X+Y)$. On the other hand there is the coproduct $\mathrm{Set}+\mathrm{Set}$. Elements are cotuples like $\langle A,X\rangle$ (is there maybe another more common notion?) or in other words a disjoint union of this two sets. I think, but I'm not sure about it, that $\langle A,X\rangle \times \langle B,Y\rangle = \langle A\times B,X\times Y\rangle $ and $\langle A,X\rangle + \langle B,Y\rangle = \langle A+ B,X+Y\rangle$. Is the internal structure of this two categories absolutely the same? So it doesn't matter whether I built tuples of sets or disjoint unions? Are they isomorphic? Or is there different behaviour?