Suppose $g: D \to C$ is a right adjoint between presentable $\infty$-categories. Then by the adjoint functor theorem, $g$ is accessible, i.e. there is a regular cardinal $\kappa$ such that both $C, D$ are $\kappa$-accessible, and $g$ preserves $\kappa$-filtered colimits. My question is then, if $C$ and $D$ are compactly generated (that is: $\omega$-accessible and closed under small colimits), can we infer that $g$ is $\omega$-accessible?
This question boils down to me not understanding the proof of Higher Topos Theory 5.4.7.7: In proving the implication '$g$ is a right adjoint, where $C, D$ are presentable' $\implies$ '$g$ is accessible', the assumptions imply that we have $f:C \to D$ a left adjoint, and that $C \simeq \operatorname{Ind}_\kappa(C')$ is the $\kappa$-filtered cocompletion of some small $\infty$-category $C'$. Compose $f$ with the Yoneda embedding $C' \to C$ to obtain a functor $f': C' \to D$. Now, by citation: Since C' is small, "there exists a regular cardinal $\tau \gg \kappa$ such that" D is $\tau$-accessible and the essential image of $f'$ is spanned by the $\tau$-compact objects of $D$. The proof then goes on to show that $g$ is $\tau$-accessible. (The statement $\tau \gg \kappa$ means by definition that $(\tau')^{\kappa'} < \tau$, whenever $\tau'< \tau$ and $\kappa' < \kappa$).
I am confused about the quoted bit: if $C$ and $D$ are both $\omega$-accessible, can we take $\tau=\kappa = \omega$?