%
\label{def:self-enrichment}
Category~$\tau$ is \textbf{self-enriched}:
it is enriched over itself
as a monoidal category
$(\tau, \times, \mathbf{1})$,
where $\times$ denotes the Cartesian product
in~$\tau$
and $\mathbf{1}$ is the terminal object.
Concretely, self-enrichment means
the following three conditions hold:
\begin{enumerate}
    \item[\textup{(SE1)}]
          \textbf{Internal Hom objects.}
          For every pair $A, B \in \mathrm{Obj}(\tau)$,
          the Hom object
          $[A,B]$
          (Definition~\ref{def:hom-object}, II.D54)
          is an object of~$\tau$.

    \item[\textup{(SE2)}]
          \textbf{Composition is a $\tau$-morphism.}
          For every triple $A, B, C \in \mathrm{Obj}(\tau)$,
          the composition map
          \[
              \circ_{A,B,C}
              \;:\;
              [B,C] \times [A,B]
              \;\longrightarrow\;
              [A,C]
          \]
          is a morphism in~$\tau$---i.e.,
          it is $\tau$-holomorphic,
          NF-addressable, tower-coherent,
          and valued in $H_\tau$.

    \item[\textup{(SE3)}]
          \textbf{Identity is a $\tau$-morphism.}
          For every $A \in \mathrm{Obj}(\tau)$,
          the identity selection
          \[
              \mathrm{id}_A
              \;:\;
              \mathbf{1}
              \;\longrightarrow\;
              [A,A]
          \]
          is a morphism in~$\tau$---i.e.,
          the identity map $\mathrm{id}_A$
          is a $\tau$-object
          sitting inside the Hom object $[A,A]$.
\end{enumerate}

\medskip\noindent
These three conditions are subject to the standard
\textbf{enriched category axioms}:
\begin{enumerate}
    \item[\textup{(EC1)}]
          \emph{Associativity.}
          The diagram
          \[
              [C,D] \times [B,C] \times [A,B]
              \;\rightrightarrows\;
              [A,D]
          \]
          commutes: composing $(h,g,f)$ as
          $(h \circ g) \circ f$ or $h \circ (g \circ f)$
          yields the same result.

    \item[\textup{(EC2)}]
          \emph{Unit laws.}
          For all $A, B$:
          $\mathrm{id}_B \circ f = f = f \circ \mathrm{id}_A$
          in $[A,B]$.
\end{enumerate}
Both (EC1) and (EC2) hold in~$\tau$
by the associativity theorem
(Theorem~\ref{thm:associativity}, II.T29,
Chapter~\ref{ch:composition-structure})
and the identity construction
(Definition~\ref{def:identity-map}, II.D40,
Chapter~\ref{ch:composition-structure}).