The c-map as a functor on certain variations of Hodge structure

We give a new manifestly natural presentation of the supergravity c-map. We achieve this by giving a more explicit description of the correspondence between projective special K\"ahler manifolds and variations of Hodge structure, and by demonstrating that the twist construction of Swann, for a certain kind of twist data, reduces to a quotient by a discrete group. We combine these two ideas by showing that variations of Hodge structure give rise to the aforementioned kind of twist data and by then applying the twist realisation of the c-map due to Macia and Swann. This extends previous results regarding the lifting of general isomorphisms along the undeformed c-map, and of infinitesimal automorphisms along the deformed c-map. We show in fact that general isomorphisms can be naturally lifted along the deformed c-map.