We know that given a topological manifold,which admit a $C^k$ structure can always makes into a $C^\infty$ structure on this manifold ,which is shown in Hirsch's differential topology book Theorem 2.10:
(a) Let $1 \leqslant r<\infty .$ Every $C^{r}$ manifold is $C^{r}$ diffeomorphic to a $ C^{\infty}$ manifold.
(b) Let $1 \leqslant r<s \leqslant \infty$. If two $C^{s}$ manifolds are $C^{r}$ diffeomorphic, they are $C^{s}$ diffeomorphic.
The question is does the set of $C^k$ structure on $M$ one to one correspond to $C^\infty$ structure on $M$?(two structures are the same if there is a corresponding isomorphism between them)
I think it's correct since: Given any $C^\infty$ structure,when forget the other orders it gives one $C^k$ altalas,if we extend it to maximal altalas give one $C^k$ structure( which is $C^k$ diffeomorphic to original smooth structure)
it's injective since any $C^\infty $ equivalent structure is also $C^k$ equivalent.By the theroem (a) this map is surjective,hence upto isomorphic ,the $C_k$ structure are one to one correspond to $C^\infty$ structure is my idea correct?
Does this means these two categories which are category of smooth manifold and $C^k$ manifold are isomorphic ?