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\label{def:enrichment-functor}
The \emph{enrichment functor}
\[
    \mathcal{F}_E : \cat{Cat}_\tau \longrightarrow \cat{Cat}_\tau
\]
acts on the category of $\tau$-categories as follows.
Given a $\tau$-category~$\mathcal{C}$
(a category enriched over~$\tau$ via~$H_\tau$),
the enrichment functor produces
$\mathcal{F}_E(\mathcal{C})$
whose data are:
\begin{enumerate}
    \item \textbf{Objects}:
          $\Obj(\mathcal{F}_E(\mathcal{C})) = \Obj(\mathcal{C})$.
          The object set does not change.
    \item \textbf{Hom objects}:
          for $A, B \in \Obj(\mathcal{C})$,
          \[
              \Hom_{\mathcal{F}_E(\mathcal{C})}(A, B)
              \;=\;
              [\Hom_{\mathcal{C}}(A, B)]_{\tau},
          \]
          the internal Hom object in~$\tau$
          associated to the Hom object of~$\mathcal{C}$.
          That is, we apply the self-enrichment of~$\tau$
          to the Hom objects of~$\mathcal{C}$.
    \item \textbf{Composition}:
          inherited from the composition morphism of~$\tau$,
          applied to the enriched Hom objects.
    \item \textbf{Identity}:
          inherited from the identity morphism of~$\tau$.
\end{enumerate}
The enrichment functor preserves the monoidal structure
and the Yoneda embedding.