What is an exponential object?

One interpretation for a power of natural numbers, such as 10^6, is that it is counting a certain collection of functions: in this case, the number of functions from a six-element set to a ten-element set. (One way to think about it is that 10^6 is the number of six-digit combinations from 000000 to 999999, and each combination corresponds to a function from \{1,\ldots,6\} to \{0,\dots,9\}: the first digit is the image of 1, the second digit is the image of 2, etc.)

In general, the number m^n is the number of functions from an n-element set to a m-element set. This is somewhat analogous to standard interpretations of other arithmetical operations:

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