The author discusses the boundedness and compactness of the products of composition and differentiation operators between Hardy spaces, namely, he finds conditions on a symbol

$\phi $ in order for the map

${C}_{\phi}Df\left(z\right)={f}^{\text{'}}\left(\phi \left(z\right)\right)$ to define a bounded (or compact) operator from

${H}^{p}$ to

${H}^{q}$, where

${H}^{p}$ stand for the Hardy space on the unit disc

$U$. His results cover the cases

$1\le p<q<\infty $ and

$2\le p=q<\infty $ and the condition, as expected, becomes a Carleson-type criterion for the measure

${\mu}_{\phi}\left(E\right)={\int}_{{\left(\phi \right)}^{-1}\left(E\right)\cap \partial U)}\frac{d\theta}{2\pi}$. In the case

$q=\infty $, it is observed that boundedness and compactness are the same and only hold for functions with

${\parallel \phi \parallel}_{\infty}<1$. A more function-theoretic characterization in terms of the Nevanlinna counting function is also provided in the case

$p=q=2$, obtaining that

${N}_{\phi}\left(w\right)=O\left(\right[log(1/{\left|w\right|\left)\right]}^{3}\left)\right]$ and

${N}_{\phi}\left(w\right)=o\left(\right[log(1/{\left|w\right|\left)\right]}^{3}\left)\right]$ corresponds to the boundedness and compactness, respectively. The proofs use standard techniques and similar results have already been known for weighted Bergman spaces.