%
\label{prop:self-enrichment}
The earned topos $\mathcal{E}_\tau$
is \textbf{enriched over itself}:
for every pair of objects $P, Q$ in $\mathcal{E}_\tau$,
the external hom-set $\Hom(P, Q)$
is recovered from the internal hom $Q^P$ by:
\[
    \boxed{%
    \Hom_{\mathcal{E}_\tau}(P,\, Q)
    \;\cong\;
    \Gamma(Q^P)
    \;:=\;
    (Q^P)(\mathbf{1}_\tau),}
\]
where $\Gamma = \Hom(\mathbf{1}_\tau, -)$
is the global sections functor.
The composition law
$Q^P \times R^Q \to R^P$
is an internal morphism in $\mathcal{E}_\tau$,
and the identity $P \to P$
corresponds to $\id \in \Gamma(P^P)$.

\textbf{Global sections recover external hom.}
By the exponential adjunction
(Theorem~\ref{thm:cartesian-closed}, I.T28)
with $A = \mathbf{1}_\tau$:
\[
    \Hom(\mathbf{1}_\tau,\, Q^P)
    \;\cong\;
    \Hom(\mathbf{1}_\tau \times P,\, Q)
    \;\cong\;
    \Hom(P,\, Q),
\]
using $\mathbf{1}_\tau \times P \cong P$
(the terminal object is the unit for products).
Since $\Hom(\mathbf{1}_\tau, Q^P) = (Q^P)(\mathbf{1}_\tau) = \Gamma(Q^P)$
by Yoneda, the first claim follows.

\textbf{Internal composition.}
Given exponentials $Q^P$, $R^Q$, and $R^P$,
the composition morphism
$\mathrm{comp} : Q^P \times R^Q \to R^P$
is the transpose of the composite:
\[
    Q^P \times R^Q \times P
    \xrightarrow{\;\id \times \mathrm{ev}\;}
    Q^P \times Q
    \xrightarrow{\;\mathrm{ev}\;}
    R,
\]
using the evaluation from $R^Q$ and then from $Q^P$
(swapping factors as needed).
This exists in $\mathcal{E}_\tau$
by Theorem~\ref{thm:cartesian-closed}.

\textbf{Internal identity.}
The identity $\id_P : P \to P$
corresponds under the adjunction
to the global section
$\id \in \Gamma(P^P) = (P^P)(\mathbf{1}_\tau)$.
In the thin setting,
$(P^P)(X) = \{*\}$ for all $X$
(since $\mathrm{supp}(P) \cap {\downarrow}X
\subseteq \mathrm{supp}(P)$ is tautological),
so $P^P$ is the terminal presheaf
and $\Gamma(P^P) = \{*\}$.