  The nature and origin of the molecular gas component located in the
circumstellar vicinity of $\eta$ Carinae are still far from being completely
understood. Here, we present Atacama Large Millimeter/Submillimeter Array
(ALMA) CO(3$-$2) observations with a high angular resolution ($\sim$0.15$''$),
and a great sensitivity that are employed to reveal the origin of this
component in $\eta$ Carinae. These observations reveal much higher velocity
($-$300 to $+$270 km s$^{-1}$) blue and redshifted molecular thermal emission
than previously reported, which we associate with the lobes of the Homunculus
Nebula, and that delineates very well the innermost contours of the red- and
blue-shifted lobes likely due by limb brightening. The inner contour of the
redshifted emission was proposed to be a {\it disrupted torus}, but here we
revealed that it is at least part of the molecular emission originated from the
lobes and/or the expanding equatorial skirt. On the other hand, closer to
systemic velocities ($\pm$100 km s$^{-1}$), the CO molecular gas traces an
inner butterfly-shaped structure that is also revealed at NIR and MIR
wavelengths as the region in which the shielded dust resides. The location and
kinematics of the molecular component indicate that this material has formed
after the different eruptions of $\eta$ Carinae.
