  We construct overconvergent dual Eichler-Shimura maps as an application of
the Higher Coleman's theory for the modular curve developed by Boxer-Pilloni.
In the road, we redo the overconvergent Eichler- Shimura map of
Andreatta-Iovita-Stevens. We give a new proof of Faltings Eichler-Shimura
decompostion via a pro\'etale dual BGG theory. Finally, we prove that the
overconvergent Poincar\'e and Serre pairings are compatible with the
overconvergent Eichler-Shimura morphisms.
