In this paper, we mainly concern with the system of the coupled nonlinear Schrödinger equation with initial data in critical spaces in dimension 3. We first obtain the global well-posedness of the solution to the coupled nonlinear Schrödinger equation with initial data in a critical space W 11 7 , 7 6 ( R 3 ) . The key is to derive a uniform bound of a modified energy E ( t ) based on a decomposition for the solution as in Dodson [Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. Rev Mat Iberoam. 2022;38:1087–1100]. In addition, inspired by Dodson [Scattering for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. International mathematics research notices; 2023], we make a new decomposition for the solution and show the scattering to the coupled nonlinear Schrödinger equation with radially symmetric initial data in a critical Besov space B 1 , 1 2 ( R 3 ) .