# A question on concrete category

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?

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.