We construct N-bright–bright and N-dark–dark soliton solutions of an integrable two coupled generalized nonlinear Schrödinger (CGNLS) equation for arbitrary values of system parameters. These solutions are more general than the reported one. While the bright–bright solitons are captured in the focusing regime of CGNLS equation, the dark–dark soliton solutions are identified in the defocusing regime. We present N-bright–bright solitons in the Gram determinant forms and prove that these determinant forms satisfy the Hirota bilinear equations.