  We prove the surprising fact that the infinity-category of stabilized
Liouville sectors is a localization of an ordinary category of stabilized
Liouville sectors and strict sectorial embeddings. From the perspective of
homotopy theory, this result continues a trend of realizing geometrically
meaningful mapping spaces through the categorically formal process of
localizing. From the symplectic viewpoint, these results allow us to reduce
highly non-trivial coherence results to much simpler verifications. For
example, we prove that the wrapped Fukaya category is coherently functorial on
stabilized Liouville sectors: Not only does a wrapped category receive a
coherent action from stabilized automorphism spaces of a Liouville sector,
spaces of sectorial embeddings map to spaces of functors between wrapped
categories in a way respecting composition actions. As a consequence, we
observe that wrapped Floer theory for sectors works in families. As we will
explain, our methods immediately establish such coherence results for most
known sectorial invariants, including Lagrangian cobordisms.
  As another application, we show that this infinity-category admits a
symmetric monoidal structure, given by direct product of underlying sectors.
The existence of this structure relies on a computation--familiar from the
foundations of factorization homology--that localizations detect certain
isotopies of smooth manifolds. Moreover, we characterize the symmetric monoidal
structure using a universal property, again producing a simple-as-possible
criterion for verifying whether invariants are both continuously and
multiplicatively coherent in a compatible way.
