  The graded Hori map has been recently introduced by Han-Mathai in the context
of T-duality as a $\mathbb{Z}$-graded transform whose homogeneous components
are the Hori-Fourier transforms in twisted cohomology associated with integral
multiples of a basic pair of T-dual closed 3-forms. We show how in the rational
homotopy theory approximation of T-duality, such a map is naturally realised as
a pull-iso-push transform, where the isomorphism part corresponds to the
canonical equivalence between the left and the right gerbes associated with a
T-duality configuration.
