%
\label{def:two-category}
The \textbf{2-category $\tau_2$} has the following data:
\begin{enumerate}
    \item[\textup{(0C)}]
          \textbf{0-cells.}
          The objects of~$\tau$:
          \[
              \mathrm{Obj}(\tau_2) \;:=\; \mathrm{Obj}(\tau).
          \]
          These are the NF-addressable objects
          from Book~I,
          organized into the primorial tower.

    \item[\textup{(1C)}]
          \textbf{1-cells.}
          For each pair of 0-cells $A, B$,
          the 1-cells from $A$ to~$B$
          are the elements of the Hom object:
          \[
              \tau_2(A,B) \;:=\; [A,B]
              \;=\; \Hom(A,B)
              \;\in\; \mathrm{Obj}(\tau).
          \]
          This is the internal hom
          of the self-enrichment (II.D54).

    \item[\textup{(2C)}]
          \textbf{2-cells.}
          For each pair of 1-cells
          $f, g \in [A,B]$,
          the 2-cells from $f$ to~$g$
          are the elements of the iterated hom:
          \[
              \tau_2(f,g) \;:=\; [f,g]
              \;\subset\; \bigl[[A,B],\,[A,B]\bigr].
          \]
          A 2-cell $\alpha \colon f \Rightarrow g$
          is a ``directed modification''
          from the morphism~$f$
          to the morphism~$g$.

    \item[\textup{(VC)}]
          \textbf{Vertical composition.}
          Given 2-cells
          $\alpha \colon f \Rightarrow g$
          and $\beta \colon g \Rightarrow h$,
          their vertical composite
          $\beta \circ_v \alpha \colon f \Rightarrow h$
          is the composition of endomorphisms
          in $\bigl[[A,B],[A,B]\bigr]$:
          \[
              \beta \circ_v \alpha
              \;:=\;
              \beta \circ \alpha
              \;\in\; \bigl[[A,B],[A,B]\bigr].
          \]
          This is ordinary composition
          in the endomorphism space,
          which is well-defined because
          $\bigl[[A,B],[A,B]\bigr]$
          is a $\tau$-object with its own composition
          (inherited from $\tau$'s enriched structure).

    \item[\textup{(HC)}]
          \textbf{Horizontal composition.}
          Given 2-cells
          $\alpha \colon f \Rightarrow f'$
          (where $f, f' \colon A \to B$)
          and $\beta \colon g \Rightarrow g'$
          (where $g, g' \colon B \to C$),
          the horizontal composite
          $\beta \circ_h \alpha \colon g \circ f \Rightarrow g' \circ f'$
          is defined by the functoriality
          of the composition map
          $\mu_{A,B,C} \colon [B,C] \times [A,B] \to [A,C]$:
          \[
              \beta \circ_h \alpha
              \;:=\;
              \mu_{A,B,C}(\beta, \alpha).
          \]
          The composition map~$\mu_{A,B,C}$
          is a $\tau$-morphism
          (it is internal to $\tau$'s enriched structure),
          so its action on 2-cells
          produces a well-defined 2-cell.

    \item[\textup{(ID)}]
          \textbf{Identities.}
          \begin{itemize}
              \item For each 0-cell~$A$:
                    the identity 1-cell
                    $\id_A \in [A,A]$.
              \item For each 1-cell $f \in [A,B]$:
                    the identity 2-cell
                    $\mathrm{id}_f \colon f \Rightarrow f$,
                    which is the identity element
                    in $[f,f] \subset [[A,B],[A,B]]$.
          \end{itemize}

    \item[\textup{(IL)}]
          \textbf{Interchange law.}
          For 2-cells
          $\alpha \colon f \Rightarrow f'$,
          $\alpha' \colon f' \Rightarrow f''$
          (both in $[[A,B],[A,B]]$),
          $\beta \colon g \Rightarrow g'$,
          $\beta' \colon g' \Rightarrow g''$
          (both in $[[B,C],[B,C]]$):
          \[
              \boxed{%
              (\beta' \circ_v \beta)
              \circ_h
              (\alpha' \circ_v \alpha)
              \;=\;
              (\beta' \circ_h \alpha')
              \circ_v
              (\beta \circ_h \alpha).}
          \]
\end{enumerate}