  We construct an explicit equivalence between the (bi)category of gl(2) webs
and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and
cobordisms. With this equivalence we can fix functoriality of every link
homology theory that factors through the Bar-Natan category. To achieve this,
we define web versions of arc algebras and their quasi-hereditary covers, which
provide strictly functorial tangle homologies. Furthermore, we construct
explicit isomorphisms between these algebras and the original ones based on
Temperley-Lieb cup diagrams. The immediate application is a strictly functorial
version of the Beliakova-Putyra-Wehrli quantization of the annular link
homology.
