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This following is excerpted from Category Theory by S. Awodey. "Theorem 1.6. Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions." Remark 1.7. This shows us what is wrong with the naive notion of a “concrete” category of sets and functions: while not every category has special sets and functions as its objects and arrows, every category is isomorphic to such a one. Thus, the only special properties such categories can possess are ones that are categorically irrelevant,..." What is the exact meaning of categorically irrelevant here?

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It means the differences would amount to not more than a renaming, as far as the algebraic axioms of the category theory defined in the book are concerned. Just as $(\{1,-1\},·)$ and $(\{0,1\},+_{\mathrm{mod} 2})$ are isomorphic groups (there is only one group of order two) and the two $\mathbb R$-metric distances of the elements of the two representations, $\|1-(-1)\|=2$ and $\|0-1\|=1$, are properties that are group theoretically irrelevant.

The excerpt alone is a little vague to me. E.g. if someone told me he works with a category of with sets as objects and functions as arrows, I'd guess but not automatically assume he implies the arrow concatenation is function concatenation. Concrete categories are of course the related term here, probably there you find examples, counter-examples and a better feel..

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    $\begingroup$ Yes, of course he means composition of functions is composition of functions. $\endgroup$ – Kevin Carlson Dec 5 '15 at 16:21

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