  Given C$^*$-algebras $A$ and $B$ and a $^*$-homomorphism $\phi:A\rightarrow
B$, we adopt the portrait of the relative $K$-theory $K_*(\phi)$ due to Karoubi
using Banach categories and Banach functors. We show that the elements of the
relative groups may be represented in a simple form. We prove the existence of
two six-term exact sequences, and we use these sequences to deduce the fact
that the relative theory is isomorphic, in a natural way, to the $K$-theory of
the mapping cone.
