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In the category of pointed sets [p.17, Awodey], can there be an arrow that points to the same set but with a different distinguishing element?

I.e., if I have a set $A=\{1,2,3\}$, can there an arrow $f: (A,1) \to (A,2)$?

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A constant map $f = 2$ for example. – Makoto Kato Oct 28 '12 at 5:08
up vote 3 down vote accepted

Sure. Any set function $f:A \to A$ with $f(1)=2$ gives such an arrow.

Two pointed sets which are the same set with a different distinguishing element aren't any more different than two totally different pointed sets (which you can obviously have functions between) — they just aren't the same object as each other.

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Awesome, just want to double check! – drozzy Oct 28 '12 at 5:24

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