Establish canonical isomorphism $Z^{Y \times X} \cong (Z^Y)^X$ for objects $X, Y$ and $Z$ from $\mathcal{AB}$ category of Abelian groups

Let $X^Y := \mathsf{Hom}_{\mathcal{AB}}(Y, X)$ be a set of all morphisms from objects $Y$ to $X$ from Abelian groups category $\mathcal{AB}$. Let $X \times Y$ be a product and $X + Y$ be a coproduct of object $X$ and $Y$.

The task is to establish canonical isomorphisms: $$Z^{Y \times X} \cong (Z^Y)^X, \quad Z^{Y+X} \cong Z^Y\times Z^X, \quad (Z \times Y)^X \cong Z^X \times Y^X.$$

I understand the task, but I dont know, how to do it specifically for $\mathcal{AB}$ (nor for any other category). It would be nice, if someone will provide me a proof please, or at least link to a source.

• $X$, $Y$, $Z$ are abelian groups, right? – Oskar Apr 8 '16 at 22:36
• The idea is to find a morphism $g: X \to Z^Y$ that sends $x \mapsto g_x$, where for $f \in Z^{X\times Y}$ we have $f(x,y) = g_x(y)$. Google "currying". – David Wheeler Apr 8 '16 at 23:30
• The first isomorphism listed is false. – Zhen Lin Apr 9 '16 at 6:41
• Instead, it holds as $Z^{Y\otimes X}\cong (Z^Y)^X$. – Berci Apr 11 '16 at 21:06