  The present research paper investigates the intricate fields of Hom-Lie
algebra and Hom-Lie coalgebra, providing a complete analysis of their key
concepts and important examples. Precisely, the paper introduces the concept of
Hom-Lie coderivation pairs and demystifies its duality with Hom-Lie derivation
pairs, inspecting pertinent facts such as representation and semi-direct
product. Furthermore, the study examines the connection between Hom-Lie,
pre-Lie, and Ass-Coder pairs with the use of crucial operators such as
commutator, Rota-Baxter operator, and endomorphism operator. Finally, the paper
concludes by presenting the construction of Hom-pre-Lie coderivation pairs
through a dual to an endomorphism operator.
