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I quite like this:
One of the important perceptions of category theory is that an arrow x : T â†’ A in a category can be regarded as an element of A defined over T. The idea is that x is a variable element of A, meaning about the same thing as the word “quantity” in such sentences as, “The quantity x2 is nonnegative”, found in older calculus books.
—Toposes, Triples and Theories, Michael Barr and Charles Wells, Version 1.1, 2002

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