We show that the set of augmentations of the Chekanov-Eliashberg algebra of a
Legendrian link underlies the structure of a unital A-infinity category. This
differs from the non-unital category constructed in [BC], but is related to it
in the same way that cohomology is related to compactly supported cohomology.
The existence of such a category was predicted by [STZ], who moreover
conjectured its equivalence to a category of sheaves on the front plane with
singular support meeting infinity in the knot. After showing that the
augmentation category forms a sheaf over the x-line, we are able to prove this
conjecture by calculating both categories on thin slices of the front plane. In
particular, we conclude that every augmentation comes from geometry.