Magnetic properties and phase composition of both MnN/CoFe (MnN at top of bilayer) and CoFe/MnN films (MnN at bottom of bilayer) bilayers through annealing at various temperatures (Ta = 300-450 °C) and then cooling to room temperature under the application of an external magnetic field of 1.5 kOe are compared. The exchange bias field (HE), the magnitude of magnetic hysteresis loop shift, of the studied films is highly related to phase composition of antiferromagnetic (AF) layer. The increase of HE with increasing Ta in the range of 300-375 °C possibly results from the improvement of magnetocrystalline anisotropy of AF related to the promoted crystallinity and stress relaxation of tetragonal face-centered θ-MnN phase. The reduction of HE at higher Ta is due to the decreased volume fraction or disappearance of θ-MnN phase and the formation of impurity phases, such as Mn4N and Mn. The induction of impurity phases is possibly related to the diffusion of part of N out of MnN phase at higher Ta. Higher HE for CoFe/MnN than MnN/CoFe at Ta = 300-375 °C might be attributed to larger amount and higher degree of stress relaxation for θ-MnN phase. For CoFe/MnN film annealed at 375 °C, the highest HE = 562 Oe is attained, and the corresponding interfacial exchange energy of 0.47 mJ/m2 in this study is comparable to that reported by Meinert et al. [Phys. Rev. B 92, 144408 (2015)].