Source: Set Theory by Kenneth Kunen.
Theorem I.12.14 ($ZFC^-$) Let $\kappa$ be an infinite ordinal. If $\mathcal{F}$ is a family of sets with $|\mathcal{F}|\leq\kappa$ and $|X|\leq\kappa$ for all $X\in\mathcal{F}$, then $|\bigcup\mathcal{F}|\leq\kappa$.
In the proof, the author goes: "...well order $\mathbf{\mathcal{S}:={}^{\kappa}(\bigcup\mathcal{F})}$, and let $g_{\alpha}$ be the least $g\in\mathcal{S}$ with $ran(g)=f(\alpha)$." Here ${}^{\kappa}(\bigcup\mathcal{F})$ denotes the set of all functions $\sigma: \kappa\to\bigcup\mathcal{F}$.
My Question: How exactly does one well order the set $\mathcal{S}$? It might be possible if $\bigcup\mathcal{F}$ is well ordered, but that is not necessary true in this case.
Any help would be greatly appreciated.
The "$C$" in Kunen's "($ZFC^-$)" means that the theorem is predicated on the axiom of choice. If you are given the axiom of choice, then any set can be well-ordered.