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  text: 'If $X$ and $Y$ are sets, the set of functions from $X$ to $Y$ (often written $X \\to Y$) is sometimes also written $Y^X$. This latter notation, which we'll call *exponential notation*, is related to the notation for finite powers of sets (e.g., $Y^3$ for the set of triples of elements of $Y$) as well as the notation of exponentiation for numbers.\n\nWithout further ado, here are some reasons this is good notation.\n\n- A function $f : X \\to Y$ can be thought of as an "$X$ wide" tuple of elements of $Y$. That is, a tuple of elements of $Y$ where the positions in the tuple are given by elements of $X$, generalizing the notation $Y^n$ which denotes the set of $n$ wide tuples of elements of $Y$. Note that if $|X| = n$, then $Y^X \\cong Y^n$.\n\n- This notion of exponentiation together with cartesian product as multiplication and disjoint union as addition satisfy the same relations as exponentiation, multiplication, and addition of natural numbers. Namely, \n\n  - $Z^{X \\times Y} \\cong (Z^X)^Y$ (this isomorphism is called currying)\n  - $Z^{X + Y} \\cong Z^X \\times Z^Y$\n  - $Z^1 \\cong Z$ (where $1$ is a one element set, since there is one function into $Z$ for every element of $Z$)\n  - $Z^0 \\cong 1$ (where $0$ is the empty set, since there is one function from the empty set to any set)\n\nMore generally, $Y^X$ is good notation for the exponential object representing $\\text{Hom}_{\\mathcal{C}}(X, Y)$ in an arbitrary cartesian closed category $\\mathcal{C}$ for the first set of reasons listed above.',
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