T-duality with $H$-flux for $2d$ $\sigma$-models

In this paper, we establish graded T-duality for $2d$ $\sigma$-models with $H$-flux after localization. This establishes the most general version of T-duality for Type II String Theory. The graded T-duality map, which we call {\bf graded Hori morphism}, is compatible with the Jacobi property of the graded fields, that was earlier studied in \cite{HM21}. Also included are some open problems/conjectures.