$\theta_s\geq_R\theta_{st}$ and hence $\theta_s\sim_R\theta_{st}$. Similarly one shows that $\theta_t\sim_L\theta_{st}$. The claim follows.

After Lemma \[lem1\], we may define the two-sided cell $\mathcal{J}$ of ${\sc\mbox{S}\hspace{1.0pt}}$ as the two-sided cell containing $\theta_s$, for all $s\in S$. As $\underline{s}$, where $s\in S$, generate $\mathbb{Z}[W]$, it follows that $\mathcal{J}$ is the unique minimal element, with respect to the two-sided order, in the set of all two-sided cells of ${\sc\mbox{S}\hspace{1.0pt}}$ different from the two-sided cell corresponding to $\theta_e$.

Let ${\sc\mbox{I}\hspace{1.0pt}}$ be the unique $2$-ideal of ${\sc\mbox{S}\hspace{1.0pt}}$ which is maximal with respect to the property that it does not contain any $\mathrm{id}_{\mathrm{F}}$, where $\mathrm{F}\in \mathcal{J}$. The $2$-category ${\sc\mbox{S}\hspace{1.0pt}}/{\sc\mbox{I}\hspace{1.0pt}}$ will be called the [*small quotient*]{} of ${\sc\mbox{S}\hspace{1.0pt}}$ and denoted by $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$. By construction, the $2$-category $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is fiat and $\mathcal{J}$-simple. It has two two-sided cells, namely, the two-sided cell corresponding to $\theta_e$ and the image of $\mathcal{J}$, which we identify with $\mathcal{J}$, abusing notation.

Generalities on simple transitive $2$-representations of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ {#s4}
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Basic combinatorics of $\mathcal{J}$ {#s4.1}
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Here we describe the basics of the Kazhdan-Lusztig combinatorics related to the two-sided cell $\mathcal{J}$. Recall Lusztig’s $\mathbf{a}$-function $\mathbf{a}:W\to \mathbb{Z}_{\geq 0}$, which is defined in [@Lu1]. This function has, in particular, the following properties: it is constant on 