The group $\mathrm{Diff}(F)$ and transition functions of a fibre bundle. Let $M$ and $F$ be differentiable manifolds, and let $F\to E\to M$ be a differentiable fibre bundle over $M$. A trivialising cover $\{(U_i,\phi_i)\,|\,i\in I\}$ of $M$ determines a set $\{t_{ij}:U_{ij}\to\mathrm{Diff}(F)\,|\,i,h\in I\}$ of transition functions satisfying the Čech cocycle conditions. In [Nakahra, Geometry, Topology and Physics, 2003], it is assumed that these transition functions actually take values in some Lie group $G$, and it is simply mentioned that the transition functions are smooth maps. I was pondering:


*

*Why are they actually smooth?

*If one considers the general case that the transition functions take values in $\mathrm{Diff}(F)$, are these functions still smooth? Is this even well-defined, i.e. is $\mathrm{Diff}(F)$ always a manifold?


Any help is much appreciated.
 A: You have to be careful when $F$ is not compact, and I don't want to be careful today, so let $F$ be a smooth compact manifold without boundary. Then $\text{Diff}(F)$ is a Frechet manifold: it's locally modeled not on Euclidean spaces, but on topological vector spaces called Frechet spaces. Here the Frechet space that's the tangent space at the identity is precisely $\mathfrak X(F)$, the space of smooth vector fields on $F$.
The point of the topology on this space is that a continuous map $M \to \text{Diff}(F)$ is canonically the same as a continuous family of diffeomorphisms $M \times F \to F$, where each $f_m$ is a diffeomorphism. So the point of the smooth structure is that a smooth map $M \to \text{Diff}(M)$ gives me a smooth family if diffeomorphisms $M \times F \to F$, and vice versa.
Now, when you pick trivializations, they are always meant to be smooth trivializations (as opposed to fiberwise smooth trivializations); and therefore the transition maps $U \times F \to U \times F$ are smooth; composing with the projection map gives us a smooth family of diffeomorphisms $U \times F \to F$. So the cocycle this gives, $U \to \text{Diff}(F)$, is smooth. So indeed the transition maps are smooth. If you only demanded the trivializations be smooth on each fiber, $U \times F \to F$ would only be a continuous family of diffeomorphisms, and $U \to \text{Diff}(F)$ would only be continuous. 
The theory of Frechet manifolds is well-developed (see Kriegl-Michor, a convenient setting for analysis), but having not read that book, I tend to try to use it as philosophy (like the above).
