\begin{abstract}
These are some minor notes and observations related to a paper by Cholak, Jockusch, and Slaman \cite{cholak:et:al:01}. In particular, if $T_1$ and $T_2$ are theories in the language of second-order arithmetic and $T_2$ is $\Pi^1_1$ conservative over $T_1$, it is not necessarily the case that every countable model of $T_1$ is an $\omega$-submodel of a countable model of $T_2$; this answers a question posed in \cite{cholak:et:al:01}. On the other hand, for $n \geq 1$, every countable $\omega$-model of $\na{I\Sigma_n}$ (resp.~$\na{B\Sigma_{n+1}}$) \emph{is} an $\omega$-submodel of a countable model of $\na{WKL_0 + I\Sigma_n}$ (resp.~$\na{WKL_0 + B\Sigma_{n+1}}$).
\end{abstract}