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let $X$ and $Y$ be sets and $Y^X$ the set of function $f:X\to Y$. How can we interpret $Y^X$ as the cartesian product $\prod_{x\in X}Y_x$ where $Y_x=Y$ for each $x\in X$?

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    Yes, that is correct.2012-02-23

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The elements in the Cartesian product $\prod_{x\in X}Y_x$ are sequences indexed by $X$ whose elements are members of $Y$, namely $\langle y_x\mid x\in X\rangle$.

Such sequence is naturally isomorphic to $\{\langle x,y_x\rangle\mid x\in X\}$, which is exactly a function from $X$ to $Y$.

This means that there is a very natural way to identify $\prod_{x\in X} Y_x$ with $Y^X$.

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    @Zhen: Also $X\times Y\times Z$ can be thrown into that mix. :-)2012-02-23