It is easy to prove that the equation $\langle \mathscr{A} \rangle X = \mathrm{atoms}^{\mathfrak{A}}\, X$ (for principal filters $X$) defines a (unique) funcoid $\mathscr{A}$ which I call quasi-atoms funcoid.
Note that as it is easy to prove $\langle \mathscr{A}^{-1} \rangle Y = \bigsqcup Y$ for every set $Y$ of ultrafilters.