Infrared imaging of the colliding-wind binary Apep has revealed a spectacular dust plume with complicated internal dynamics that challenges standard colliding-wind binary physics . Such challenges can be potentially resolved if a rapidly-rotating Wolf-Rayet star is located at the heart of the system , implicating Apep as a Galactic progenitor system to long-duration gamma-ray bursts . One of the difficulties in interpreting the dynamics of Apep is that the spectral composition of the stars in the system was unclear . Here we present visual to near-infrared spectra that demonstrate that the central component of Apep is composed of two classical Wolf-Rayet stars of carbon- ( WC8 ) and nitrogen-sequence ( WN4-6b ) subtypes . We argue that such an assignment represents the strongest case of a classical WR+WR binary system in the Milky Way . The terminal line-of-sight wind velocities of the WC8 and WN4-6b stars are measured to be 2100 \pm 200 and 3500 \pm 100 km s ^ { -1 } , respectively . If the mass-loss rate of the two stars are typical for their spectral class , the momentum ratio of the colliding winds is expected to be \approx 0.4 . Since the expansion velocity of the dust plume is significantly smaller than either of the measured terminal velocities , we explore the suggestion that one of the Wolf-Rayet winds is anisotropic . We can recover a shock-compressed wind velocity consistent with the observed dust expansion velocity if the WC8 star produces a significantly slow equatorial wind with a velocity of \approx 530 km s ^ { -1 } . Such slow wind speeds can be driven by near-critical rotation of a Wolf-Rayet star .