K1 of a ring as relative K1 group

Let be a ring
Then there exists a natural isomorphism

for the

1. relative K1-Group functor from Pairs of Rings
   
2. k1 functor from Ring
   

Note that - either by explicite computation or by abstract nonsense (here the diagram is an equivalence to the the constant diagram from a point, equivalence of categories as initial functor)

Thus we get natural isomorphism

(cf infinite general linear group functor, infinite elementary linear group functor)

which induces a natural isomorphism

Then applictaion of the cokernel cokernel functor preserves the isomorphism