  If $b$ is an inner function, then composition with $b$ induces an
endomorphism, $\beta$, of $L^\infty(\mathbb{T})$ that leaves
$H^\infty(\mathbb{T})$ invariant. We investigate the structure of the
endomorphisms of $B(L^2(\mathbb{T}))$ and $B(H^2(\mathbb{T}))$ that implement
$\beta$ through the representations of $L^\infty(\mathbb{T})$ and
$H^\infty(\mathbb{T})$ in terms of multiplication operators on
$L^2(\mathbb{T})$ and $H^2(\mathbb{T})$. Our analysis, which is based on work
of R. Rochberg and J. McDonald, will wind its way through the theory of
composition operators on spaces of analytic functions to recent work on Cuntz
families of isometries and Hilbert $C^*$-modules.
