%
\label{def:cat-tau}
The \textbf{earned category} $\mathrm{Cat}_\tau$
is the category defined by:
\begin{enumerate}
    \item \textbf{Objects}: $\mathrm{Ob}(\mathrm{Cat}_\tau) := \mathrm{Ob}(\tau)$,
          the $\tau$-index set ---
          the vertices of the primorial tower
          determined by the 9~axioms.
    \item \textbf{Morphisms}: for objects $A, B \in \mathrm{Ob}(\tau)$,
          the hom-set is:
          \[
              \boxed{%
              \mathrm{Hom}_{\mathrm{Cat}_\tau}(A, B)
              \;:=\;
              \bigl\{\,
                  \alpha = [\pi]_{\mathrm{NF}}
                  : T_\alpha \in \mathrm{HolFun},\;
                  \mathrm{src}(\alpha) = A,\;
                  \mathrm{tgt}(\alpha) = B
              \,\bigr\},}
          \]
          where $T_\alpha$ is the HolFun
          carried by the $\tau$-arrow $\alpha$
          (Definition~\ref{def:tau-arrow}, I.D50),
          and $\mathrm{src}$, $\mathrm{tgt}$
          assign source and target objects
          via the HolMap structure
          (Definition~\ref{def:holmap}, I.D48).
    \item \textbf{Composition}:
          for $\alpha \in \mathrm{Hom}(A, B)$
          and $\beta \in \mathrm{Hom}(B, C)$,
          the composite is:
          \[
              \beta \circ \alpha
              := [\pi_\beta \circ \pi_\alpha]_{\mathrm{NF}},
          \]
          which lies in $\mathrm{Hom}(A, C)$
          by the composition closure of HolFun
          (Theorem~\ref{thm:composition-closure}, I.T20).
    \item \textbf{Identity}: for each object $A \in \mathrm{Ob}(\tau)$,
          the identity arrow is:
          \[
              \mathrm{id}_A := [\mathrm{id}_\tau]_{\mathrm{NF}},
          \]
          the NF-equivalence class of the identity transformer.
\end{enumerate}