The symbol ${\approx}$ in the last postulate is an abbreviation. For subsets $Y$ of ${\mathfrak G}$, equinumerosity can be defined as the existence of a set of pairs, $f=\{\{y,u\},\{y',u'\},\dots\}$, which puts elements $y\in Y$ uniquely in correspondence with $u\in{\mathfrak G}$:

I reverse-engineered this from the metamath page, so it needs to be checked

$Y\ {\approx}\ {\mathfrak G}\equiv\exists f. \forall x.\left((x \in Y) \implies \exists!u.\ \{x, u\} \in f\right) \land \left((x \in {\mathfrak G}\setminus Y) \implies \exists(y \in Y).\ \{y,x\} \in f\right)$

The first clause says $f$ corresponds to a function $F$ and moreover implies $F(b) \in Y \implies F(F(b))=b$. The second says $F$ is surjective into ${\mathfrak G}\setminus Y$.

strangely, to me, the metamath defintion doesn't include transitivity in an obvious way, and I can't see it… I added it here anyway, since it's part of any other definition I've seen.

Wikipedia: Grothendieck universe, Universe (Mathematics)

They have the corresponding axiom in pure first order logic language and 6 shorter version on metamath.

Metamath: grothprim (long), axgroth5 (the one above)

nLab: Grothendieck universe

Bijective function, Power set