%
\label{prop:thin-category}
$\mathrm{Cat}_\tau$ is a \textbf{thin category}:
for all objects $A, B \in \mathrm{Ob}(\tau)$,
\[
    \boxed{%
    \bigl|\,\mathrm{Hom}_{\mathrm{Cat}_\tau}(A, B)\,\bigr|
    \;\leq\; 1.}
\]
That is: if $\alpha, \beta \in \mathrm{Hom}(A, B)$,
then $\alpha = \beta$.

Let $\alpha, \beta \in \mathrm{Hom}(A, B)$.
Each arrow carries a unique HolFun:
$T_\alpha, T_\beta \in \mathrm{HolFun}$.
Both are $\tau$-holomorphic functions
with the same source $A$ and target $B$.

Since $T_\alpha$ and $T_\beta$
share the same source rooting,
they act on the same set of omega-tails.
At the source object $A$,
both functions agree on all omega-tails of depth $1$
(the shallowest primorial stage) ---
because the source rooting determines
the depth-$1$ behavior,
and both functions are tower-coherent
from the same source $A$.

Therefore there exists a depth $d_0 = 1$
at which $T_\alpha$ and $T_\beta$ agree.
By the $\tau$-Identity Theorem
(Theorem~\ref{thm:tau-identity}, I.T21):
\[
    T_\alpha \sim_{d_0} T_\beta
    \;\;\Longrightarrow\;\;
    T_\alpha = T_\beta.
\]
Since $T_\alpha = T_\beta$ as HolFuns,
we have $\alpha = \beta$ as $\tau$-arrows.