  We define a family of formal Khovanov brackets of a colored link depending on
two parameters. The isomorphism classes of these brackets are invariants of
framed colored links. The Bar-Natan functors applied to these brackets produce
Khovanov and Lee homology theories categorifying the colored Jones polynomial.
Further, we study conditions under which framed colored link cobordisms induce
chain transformations between our formal brackets. We conjecture that, for
special choice of parameters, Khovanov and Lee homology theories of colored
links are functorial (up to sign). Finally, we extend the Rasmussen invariant
to links and give examples, where this invariant is a stronger obstruction to
sliceness than the multivariable Levine-Tristram signature.
