Conjugate objects in C*-categories and conjugate superselection sectors

The starting point of this work is the operation of conjugation on finite-dimensional objects in strict tensor C$^*$-category $\mathbf{end}(\mathscr{A})$ of localized endomorphisms $\rho$ of the quasilocal C$^*$-algebra $\mathscr{A}$ of observables [1]. The equivalence classes of irreducible, translation covariant localized endomorphisms are the superselection sectors of $\mathscr{A}$, where the tensor product is the composition of endomorphisms. The intertwiner space $(\rho,\sigma)$ between $\rho,\sigma \in \mathbf{end}(\mathscr{A})$
$$(\rho,\sigma) = t \in \mathscr{A} : t\rho(a) = \sigma(a)t ,\forall a \in \mathscr{A}.$$
The set of superselection sectors is endowed with a natural conjugation. The $ \bar{\rho} \in \mathbf{end}(\mathscr{A})$ is a conjugate of $\rho$ iff the conjugate equation holds true and there exist multiples of isometries $r \in (\i, \bar{\rho}\rho)$ and $\bar{r} \in ({\i}, \rho\bar{\rho})$ [1].
In [2] Doplicher and Roberts construct the field net as crossed product of the observable algebras by a semigroup of endomorphisms. In this sense the crossed product contains both $\mathscr{A}$ and the Cuntz algebra $\mathscr{O}_d$ generated by multiplet of isometries $\{\psi_i \}^d_{i=1}$. Therefore it is possible to formulate our constructions in terms of isometries $\psi_i$. Here we will consider the category generated by one (canonical) endomorphism $\rho=\sum_{i=1}^{3}\psi_ia\psi_i^*$ (with dimension $d = 3$.)
Proposition. Let, $r=\sum_{i=1}^{3}\hat{\psi_i}\psi_i$ and $\bar{r}=\sum_{i=1}^{3}\psi_i\hat{\psi_i}$. Then $\bar{\rho}=\sum_{i=1}^{3}\hat{\psi_i}a\hat{\psi}_i^*$, is such that $r\i(a) = \bar{\rho}\rho(a)r (a \in \mathscr{A})$ and the conjugate equation is satisfyed.
Here $\hat{\psi}_i=\frac{1}{2!}\sum_{p \in \mathbb{P}_3(i)} sign(p)\psi_{p(2)}\psi_{p(3)}$, $\mathbb{P}_3(i)$ is the subset in $\mathbb{P}_3$ with $p(1) = i$. If such $\bar{\rho}$ exist, then we can find the left inverse $\varphi$ to be set $\varphi_{\sigma,\tau} : (\rho\sigma,\rho\tau)\to(\sigma,\tau)$, where $\rho,\sigma,\tau \in \mathbf{end}(\mathscr{A})$. Let, $x \in (\rho\bar{\rho}, \bar{\rho}\rho)$. Then a simple computation shows, that $\phi_{\bar{\rho},\bar{\rho}}(x)=\sum_j\psi_j^*x\psi_j\in(\bar{\rho},\bar{\rho})$.
If $A^{\rho}_{d=3}$ is the projector on to the antisymmetric subspace in $(\rho^3,\rho^3)$, then a computation shows that $\phi(A^{\rho}_{d=3})=\frac{1}{3}\sum_{i,j,k}\psi_k^*\psi_i\hat{\psi}_i\hat{\psi}_j^*\psi_j^*\psi_k, A^{\rho}_{3}=\frac{1}{3}\sum_{i,j}\psi_i\hat{\psi}_i\hat{\psi}_j^*\psi_j^*$.