\label{def:enrichment-functor-e01}
The \emph{enrichment functor} is a faithful, tower-coherent functor
\begin{equation}
\operatorname{Enr}_{01} : \operatorname{Cat}_{\tau}(\Elayer{0})
\;\longrightarrow\;
\operatorname{Cat}_{\tau}(\Elayer{1})
\label{eq:ch44-enr01}
\end{equation}
defined on the category of $\tau$-objects at enrichment level~$\Elayer{0}$. It acts as follows.
\begin{enumerate}
\item\emph{(On objects.)}
An $\Elayer{0}$-object $X = (U, \phi, \operatorname{NF}(X))$---a clopen cylinder domain~$U$, a presheaf section~$\phi$, and a normal-form address---is sent to the $\Elayer{1}$-object
\[
\operatorname{Enr}_{01}(X) \;=\; \bigl(U,\; \phi^{\jj},\; \operatorname{NF}(X),\; \boldsymbol{S}(X),\; \Delta(X, \cdot)\bigr),
\]
where $\phi^{\jj}$ is the split-complex extension of~$\phi$ (taking values in $H_\tau = \mathbb{Z}[\jj] / (\jj^2 - 1)$), $\boldsymbol{S}(X) \in \{A, B, C, D, \omega\}$ is the sector assignment from the $4{+}1$ decomposition (Definition~\ref{def:four-plus-one-decomposition}, Ch.~10), and $\Delta(X, \cdot)$ is the defect functional (Definition~\ref{def:defect-functional}, Ch.~35) evaluated on~$X$.

\item\emph{(On morphisms.)}
A tower-coherent morphism $f : X \to Y$ in $\operatorname{Cat}_{\tau}(\Elayer{0})$ is sent to the $\Elayer{1}$-morphism $\operatorname{Enr}_{01}(f)$ that preserves the split-complex extension, respects the sector assignment (i.e., $\boldsymbol{S}(f(X)) = \boldsymbol{S}(X)$ when $f$ is sector-preserving), and satisfies the defect inequality $\Delta(Y, n) \leq \Delta(X, n)$ for all primorial depths $n$ at which $f$ is defined.
\end{enumerate}
The functor preserves tower coherence (I.T18) and spectral naturality (I.T40): the diagrams that commute at~$\Elayer{0}$ continue to commute at~$\Elayer{1}$.