Products in a category

Question

Let $C$ be a category and let A,B,C be three objects of $C$..Should the products
$Π$(B,C), $Π$(A,B,C), $Π$(Α,$Π$(B,C)) exist, is it true in general that $Π$(A,B,C) and $Π$(Α,$Π$(B,C)) are isomorphic?? Or does the statement holds for certain types of categories, e.g. Additive ??

Answer 1

Yes, these must be isomorphic. You can prove this by the Yoneda lemma. Let $X = \prod (A,B,C)$ and $Y = \prod (A, \prod(B,C))$. Then by universal property, we have natural isomorphisms of sets$\DeclareMathOperator{\Hom}{Hom}$

$$\Hom(Z,X) \cong \Hom(Z,A) \times \Hom(Z,B) \times \Hom(Z,C)\\ \cong \Hom(Z,A) \times \Hom(Z,\prod(B,C)) \cong \Hom(Z,Y)$$

(these are natural in $Z$). This shows that $\Hom({-},X) \cong \Hom({-},Y)$, so $X \cong Y$.