  We investigate a construction providing pairs of Calabi-Yau varieties
described as zero loci of pushforwards of a hyperplane section on a roof as
described by Kanemitsu. We discuss the implications of such construction at the
level of Hodge equivalence, derived equivalence and $\mathbb L$-equivalence.
For the case of $K3$ surfaces, we provide alternative interpretations for the
Fourier-Mukai duality in the family of $K3$ surfaces of degree 12 of Mukai. In
all these constructions the derived equivalence lifts to an equivalence of
matrix factorizations categories.
