For a holomorphic self-map $\varphi$ of the unit disk of the
complex plane, the compactness of the composition operator
$C_{\varphi}(f) = f\circ \varphi$ on the Hardy spaces is known
to be equivalent to the various function theoretic conditions
on $\varphi$, such as Shapiro's Nevanlinna counting function
condition, MacCluer's Carleson measure condition, Sarason condition
and Yanagihara-Nakamura condition, etc. A direct function-theoretic
proof of Shapiro's condition and Sarason's condition was recently
given by Cima and Matheson. We give another direct function-theoretic
proof of the equivalence of these conditions by use of Stanton's
integral formula.