Foreman (2013) proved a Duality Theorem which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of ω₁ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie (2010)). As an application we prove that if ω₁ is generically strong, then it remains so after adding any number of Cohen subsets of ω₁; however many other ω₁-closed posets-such as Col(ω₁,ω₂)-can destroy the generic strongness of ω₁. This generalizes some results of Gitik-Shelah (1989) about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than ω₁.