In proximal causal inference framework, the identification of average treatment effect (ATE) depends on finding the bridge functions. The bridge functions are functions about proxy variables used in the proximal standardization formulae. They are the solutions to two Fredholm integral equations of the first kind, whose existence is determined by Picard's conditions about the singular systems of two conditional expectation operators. However, since singular systems required by Picard's conditions are hard to determine, it is an extremely tough task to solve the bridge functions directly from the integral equations. Therefore, people turn to find estimators of the bridge functions. Many literatures have provided approaches to the estimators under certain assumptions although which inevitably restrict the feasibility of their application. In this thesis, we propose a kernel embedded estimator for the treatment confounding bridge function ($q$-bridge function) based on a dual kernel embedding method, under the assumption that there exist at least one bounded continuous $q$-bridge function for each treatment. In addition, we show the consistency of the $q$-bridge function estimator and give a consistent ATE estimator based on the proximal inverse probability weighted estimator.