Can pointed sets category have an arrow that points to the same set?

Question

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 $f: (A,1) \to (A,2)$?

Answer 1

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.