Weak Rickart and dual weak Rickart objects in abelian categories: transfer via functors

Weak relative Rickart objects generalize relative Rickart objects in abelian categories. We study how such a property is preserved or reflected by fully faithful functors and adjoint pairs of functors. Various consequences are obtained for (co)reflective subcategories, adjoint triples of functors and endomorphism rings of modules. In particular, for a right H-module M with endomorphism ring S, we prove that if M is a weak self-Rickart right R-module, then 5 is a weak self-Rickart right S-module, while the converse holds provided M is a flat left S-module or M is a k-local-retractable right R-module.