I am currently reading Quantum Invariants of Knots and 3-Manifolds by Turaev, and I am having a hard time understanding a statement made on page 120. He is explaining the property of space-structures, functors $U$ between some category of topological spaces and their homeomorphisms to the category of sets and bijections. He defines a structure as involutive if
- the canonical mapping $U(X) \times U(Y) \rightarrow U(X\sqcup Y)$ is equivariant (where $X$ and $Y$ are topological spaces and $\sqcup$ is a disjoint union)
- for any homeomorphism $f$ between spaces $X$ and $Y$, the induced bijection $U(f)$ is equivariant
He then goes on to what he considers an "obvious" example of the category of n-dimensional topological manifolds where $U$ takes $X$ to the set of all smooth structures on $X$, and claims that this functor is involutive without much explanation. Why is this example of a space structure so clearly involutive? To what extent is the above definition of involutive similar to the traditional one of $f(f(X))=X$?