I am currently reading Turaev's paper Axioms for topological quantum field theory.
In couple of place, there is a paraphrase "... is natural with respect to $\mathfrak{U}$-homeomorphism" and I don't really understand what does it mean. One is in the definition of self-dual modular functor.
It says system of bilinear pairing is natural with respect to $\mathfrak{U}$-homeomorphism and the other is in the definition of cobordism. It says gluing is natural with respect to $\mathfrak{U}$-homeomorphism. It seem like that paraphrase is something I should get used to before jumping into Turaev's papers.
If so, a book suggestion would be very helpful. If not, can someone explain what Turaev means by the paraphrase.