Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is a property that the weak operator topology can detect, if $A$ were Abelian, then $A^{**}$ would be as well.
Now, in the Abelian case, $A\cong C_0(\Omega)$ and $A^{**}\cong C(\dot\Omega)$ (with $\dot\Omega$ necessarily a totally disconnected compact Hausdorff space). What does the functor that takes $\Omega$ to $\dot\Omega$ look like? Is it adjoint to the forgetful functor from the category of totally disconnected Hausdorff spaces to LCHaus? Is there some explicit description of $\dot\Omega$ based on $\Omega$?