# How does one prove this bijection of hom-sets?

If $X,Y$ and $Z$ are sets, how do you prove the bijection $$\text{Hom}(Y\times X,Z)\cong \text{Hom}(Y,\text{Hom}(X,Z))\;\;\;\;?$$ This is a specification of an "adjunction" in Category theory. I wonder if it has any applications in the form stated above.

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Related: currying. –  dtldarek Mar 11 '14 at 8:43
Isn't this the same question with this‌​? –  frabala Mar 11 '14 at 14:40

• Any map $\psi\colon Y\times X\to Z$ gives you a map \begin{align} \tilde \psi\colon Y &\longrightarrow \operatorname{Hom}(X,Z), \\ y &\longmapsto \psi(y,\,\cdot\,). \end{align}
• Any map $\tilde\varphi\colon Y \to \operatorname{Hom}(X,Z)$ gives you a map \begin{align} \varphi\colon X\times Y &\longrightarrow Z, \\ (x,y) &\longmapsto \big((\tilde\varphi)(y)\big)(x). \end{align}