  We give a functorial construction of the genus zero chiral algebras of class
$\mathcal{S}$, that is, the vertex algebras corresponding to the theory of
class $\mathcal{S}$ associated with genus zero pointed Riemann surfaces via the
4d/2d duality discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van
Rees in physics. We show that there is a unique family of vertex algebras
satisfying the required conditions and show that they are all simple and
conformal. In fact, our construction works for any complex semisimple group G
that is not necessarily simply laced. Furthermore, we show that the associated
varieties of these vertex algebras are exactly the genus zero Moore-Tachikawa
symplectic varieties that have been recently constructed by Braverman,
Finkelberg and Nakajima using the geometry of the affine Grassmannian for the
Langlands dual group.
