{\mathrm{End}}_{\mathcal{S}}({\mathbb{1}})$ is computed by $$\sigma_I=\dim(I).$$ Indeed, since ${\mathcal{S}}$ is ribbon with trivial twist, we have ${\mathrm{tr}}(\tau_{I,I})=\dim(I)$ and so $$\sigma_I={\left\langle\tau_{I,I}\right\rangle}_{I \otimes I}=\dim(I \otimes I)\, {\mathrm{tr}}(\tau_{I,I})=\dim(I)^{3}=\dim(I).$$

Example of graded vector spaces {#sect-ex-G-vect}
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Let $G$ be a group with unit element $1$. Recall that a vector space $V$ over a field ${\Bbbk}$ is $G$-graded if it is decomposed as a direct sum $$U= \bigoplus_{g \in G} V_g$$ of vector subspaces labeled by $g\in G$. For $g \in G$, the elements of $V_g$ are said to be homogeneous of degree $g$. For example, each $h\in G$ gives rise to a $G$-graded [$\Bbbk$-]{}vector space ${\Bbbk}_h $ where $({\Bbbk}_h)_h={\Bbbk}$ and $({\Bbbk}_h)_g=0$ for all $g\in G\setminus \{h\}$. We denote by $G{\,\mbox{-}\,}\mathrm{Vect}_{\Bbbk}$ the category of $G$-graded [$\Bbbk$-]{}vector spaces and [$\Bbbk$-]{}linear grading-preserving homomorphisms. We endow this category with the monoidal product defined on objects by $$U \otimes V=\bigoplus_{g \in G} \, (U \otimes V)_g \quad \text{where} \quad (U \otimes V)_g=\bigoplus_{\substack{h,j \in G \\ \, hj=g}} U_h \otimes_{\Bbbk}V_j ,$$ and on morphisms by $f \otimes g=f \otimes_{\Bbbk}g$. For example, ${\Bbbk}_g \otimes {\Bbbk}_h \simeq {\Bbbk}_{gh}$ for all $g,h \in G$. Then $G{\,\mbox{-}\,}\mathrm{Vect}_{\Bbbk}$ is a monoidal category with unit object ${\mathbb{1}}={\Bbbk}_1$. The group of isomorphism classes of invertible objects of $G{\,\mbox{-}\,}\mathrm{Vect}_{\Bbbk}$ 