In the present article we aim to broaden the consideration of background geometry of manifolds/bundles arising in heterotic compactifications with an aim towards extending the validity and understanding of heterotic/F-theory duality. In particular, we will focus on elliptically fibered Calabi-Yau geometries arising in heterotic theories in the context of the so-called Fourier Mukai transforms of vector bundles on elliptically fibered manifolds. The duality between the Heterotic and F-theory is a powerful tool in gaining more insights into F-theory description of low-energy chiral multiplets. We propose a generalization of heterotic/F-theory duality and in order to complete the translation, the dictionary of the heterotic/F-theory duality has to be refined in some aspects. The precise map of spectral surface and complex structure moduli is obtained, and with the map, we find that divisors specifying the line bundles correspond precisely to codimension singularities in F-theory.