String nets colored by a modular fusion category C can be used to construct the correlators for rational CFTs with chiral data given by C, for arbitrary world sheets including physical boundaries as well as line and point defects. The data involved in the construction comprise a pivotal bicategory Fr(C) of (simple special symmetric) Frobenius algebras.
One can define Fr(C)-colored string nets in such a way that the local rules for defects which relate world sheets sharing the same correlator amount to mapping class group intertwiners -- which we call "universal correlators" -- between the string-net spaces for Fr(C) and those for C.
These structures fit neatly into the framework of double categories: the open-closed string-net modular functors extend canonically to symmetric monoidal double functors, and the universal correlators give rise to monoidal vertical transformations.