Lifting of cohomology and unobstructedness of certain holomorphic maps

Let f be a holomorphic mapping between compact complex manifolds. We give a criterion for f to have unobstructed deformations, i.e. for the local moduli space of f to be smooth: this says, roughly speaking, that the group of infinitesimal deformations of f, when viewed as a functor, itself satisfies a natural lifting property with respect to infinitesimal deformations. This lifting property is satisfied e.g. whenever the group in question admits a 'topological' or Hodge-theoretic interpretation, and we give a number of examples, mainly involving Calabi-Yau manifolds, where that is the case.