\label{thm:base-change-transfer}
Let $f\colon S_1 \to S_2$ be a tower-coherent sector morphism in $\operatorname{Cat}_{\T}(\Elayer{1})$.
\begin{enumerate}
\item\emph{(Base change.)} The base-change map $\operatorname{BC}_f\colon \operatorname{Bdy}_{S_1}(\Elayer{0}) \to \operatorname{Bdy}_{S_2}(\Elayer{1})$ defined by $\operatorname{BC}_f(\beta_0) = f^{\mathrm{bdy}}(\operatorname{Enr}_{01}(\beta_0))$ is natural with respect to the enrichment tower: for every tower-coherent map $g\colon S_0 \to S_1$ at $\Elayer{0}$, the diagram
\begin{equation}
\begin{tikzcd}
\operatorname{Bdy}_{S_0}(\Elayer{0}) \arrow[r, "g^{\mathrm{bdy}}"] \arrow[d, "\operatorname{BC}_{f \circ g}"'] & \operatorname{Bdy}_{S_1}(\Elayer{0}) \arrow[d, "\operatorname{BC}_f"] \\
\operatorname{Bdy}_{S_2}(\Elayer{1}) \arrow[r, equal] & \operatorname{Bdy}_{S_2}(\Elayer{1})
\end{tikzcd}
\label{eq:ch49-base-change-naturality}
\end{equation}
commutes, i.e., $\operatorname{BC}_f \circ g^{\mathrm{bdy}} = \operatorname{BC}_{f \circ g}$.

\item\emph{(Transfer.)} The transfer map $\operatorname{Trf}_f\colon \Delta_{S_1}(\cdot, n) \to \Delta_{S_2}(\cdot, n)$ is natural with respect to the defect tower: for every primorial depth~$n$ and every tower-coherent map $g\colon S_0 \to S_1$,
\begin{equation}
\operatorname{Trf}_f \circ \operatorname{Trf}_g \;=\; \operatorname{Trf}_{f \circ g}.
\label{eq:ch49-transfer-functoriality}
\end{equation}
Moreover, transfer is contractive: $\Delta_{S_2}(f(X), n) \leq \Delta_{S_1}(X, n)$ for all~$n$.
\end{enumerate}