Bott Periodicity for C*-Algebras as weak homotopy equivalence

Question

Let $A$ be a unital $C^*$-Algebra. If I understand correctly we can consider $GL(A):=colim_n GL_n(A)$ top be a pointed space where the base point ist the identity. Now, if $\Omega GL(A)$ denotes the loop space then the (I think) bott periodicity says that there is a weak homotopy equivalence $$ GL(A) \simeq \Omega^2 GL(A) $$ that is natural in $A$ for unital $*$-homomorphisms. I've come across this a number of times, however I have never seen a proof when it wa mentioned. The only proof I can find is this paper from the 60s by R. Wood and I am not entirely sure that this is what he proves. So my question is: Do I understand this theorem correctly? Does the paper I've linked here really prove exactly this (including the naturality) and are there other sourced where this might be proven. Thank you.