So let us fix an arbitrary set $X\in$ Set, I want to find the universal arrow from $X$ to the forgetful functor $F:$ Set$_*\rightarrow$ Set, where Set$_*$ is the category of pointed sets.
By definition, I would want to produce an ordered pair $<\Omega, \phi>$, $\phi:X \rightarrow F\Omega$,
such that for any given pair of $<\Theta, \psi>$, there is an unique (base-preserving) function $f : \Omega \rightarrow \Theta$ where $\psi = Ff\ \circ \phi$
So firstly I want to produce a pointed set $\Omega$ (that depends on $X$), I had tried to use AC to well order $X$ and choose the least element as my base, but this does not guarantee an unique $f$ between my $\Omega$ and another arbitrary pointed set $\Theta$.
So how to do go about producing the required ordered pair (the universal arrow)? Any hints or insight is deeply appreciated.
Cheers and thanks