We construct the Eichler-Shimura morphisms for families of overconvergent modular forms via Scholze's theory of pro-etale site, as well as the Hodge-Tate period maps on modular curves of infinite level. We follow some of the main ideas in the work of Andreatta-Iovita-Stevens. In particular, we reprove the main result in their paper. Since we work entirely on the generic fiber of the modular curve, log structures will not be needed if we only consider the Eichler-Shimura morphism for cusp forms.  Moreover, the well-established theory of the Hodge-Tate period map for Shimura varieties of Hodge type may allow us to generalize the construction to more general Shimura varieties. This is a joint work with Hansheng Diao.