%
\label{lem:naturality-cylinder}
Let $f$ be an $\omega$-germ transformer.
Then for every stage~$k$ and every point $x \in \tau^3$:
\[
    \boxed{f\bigl(C_k(x)\bigr)
    \;\subseteq\;
    C_k\bigl(f(x)\bigr),}
\]
where $C_k(x)$ denotes the stage-$k$ cylinder at~$x$
(Definition~\ref{def:stage-k-cylinder}, Chapter~\ref{ch:cylinder-domains}).
That is, $f$ maps the stage-$k$ cylinder at~$x$
into the stage-$k$ cylinder at~$f(x)$.

Let $y \in C_k(x)$.
By the definition of stage-$k$ cylinder (II.D10),
this means $\pi_k(y) = \pi_k(x)$:
the points $x$ and~$y$ agree at stage~$k$
of the primorial tower.

By tower coherence of~$f$
(the naturality condition of Notation~\ref{not:ch11-omega-germ}):
\[
    \pi_k\bigl(f(y)\bigr)
    \;=\;
    f_k\bigl(\pi_k(y)\bigr)
    \;=\;
    f_k\bigl(\pi_k(x)\bigr)
    \;=\;
    \pi_k\bigl(f(x)\bigr).
\]
Hence $f(y) \in C_k(f(x))$.
Since $y \in C_k(x)$ was arbitrary,
$f(C_k(x)) \subseteq C_k(f(x))$.