I'll define a "partially wrapped" Fukaya category associated to a Liouville domain with "stops", which are Liouville hypersurfaces of the boundary. This category is closely related to Legendrian contact homology [Ekholm--Lekili, 1701.01284]. I'll explain how certain functors between partially wrapped Fukaya categories, when they exist, can give strong constraints on the structure of Legendrian contact homology. This existence can in some cases be inferred from mirror symmetry, and in general it conjecturally follows from the existence of certain positive loops.