We report a study on a hybrid mode-locked fiber laser with two saturable absorbers: slow and fast, integrated in a single device. Amorphous antimony telluride (Sb(2)Te(3)) layer was deposited on side-polished fiber to form the slow saturable absorber due to the third order nonlinear susceptibility of Sb(2)Te(3). Additionally, an unsymmetrical design of the device causes polarization-dependent losses and together with polarization controller allows to use a nonlinear polarization evolution to form the artificial fast saturable absorber. Sub-200 fs soliton pulses with 0.27 nJ of pulse energy were generated in the hybrid mode-locked Er-doped fiber laser. Differences in the dynamics of mode-locked laser are further investigated with the use of slow and fast saturable absorbers solely, and compared with the hybrid device. Joint operation of two saturable absorbers enhances the laser performance and stability. The conducted experiments allowed to define roles of each mechanism on the pulse shaping in the laser cavity.