  We study the exact extremal orders of compositions $f(g(n))$ of certain
arithmetical functions, including the functions $\sigma(n)$, $\phi(n)$,
$\sigma^*(n)$ and $\phi^*(n)$, representing the sum of divisors of $n$, Euler's
function and their unitary analogues, respectively. Our results complete,
generalize and refine known results.
