  We continue the mathematical development of the open/closed correspondence
proposed by Mayr and Lerche-Mayr. Given an open geometry on a toric Calabi-Yau
3-orbifold $\mathcal{X}$ relative to a framed Aganagic-Vafa outer brane
$(\mathcal{L},f)$, we construct a toric Calabi-Yau 4-orbifold
$\widetilde{\mathcal{X}}$ and identify its genus-zero Gromov-Witten invariants
with the disk invariants of $(\mathcal{X},\mathcal{L},f)$, generalizing prior
work of the authors in the smooth case. We then upgrade the correspondence to
the level of generating functions, and prove that the disk function of
$(\mathcal{X},\mathcal{L},f)$ can be recovered from the equivariant
$J$-function of $\widetilde{\mathcal{X}}$. We further establish a B-model
correspondence that retrieves the B-model disk function of
$(\mathcal{X},\mathcal{L},f)$ from the equivariant $I$-function of
$\widetilde{\mathcal{X}}$, and show that the correspondences are compatible
with mirror symmetry in both the open and closed sectors.
