  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.
