an isomorphism between $\mathbb{Z}[W]$ and the Grothendieck ring of the monoidal category of Soergel bimodules for $(W,S,V)$. This isomorphism sends $\underline{w}$ to the class of $B_w$, see [@So2; @E; @EW].

Let $\mathcal{C}$ be a small category equivalent to the category $\mathtt{C}\text{-}\mathrm{mod}$. Define the $2$-category ${\sc\mbox{S}\hspace{1.0pt}}={\sc\mbox{S}\hspace{1.0pt}}(W,S,V,\mathcal{C})$ of [*Soergel bimodules*]{} associated to $(W,S,V,\mathcal{C})$ as follows:

-   ${\sc\mbox{S}\hspace{1.0pt}}$ has one object $\mathtt{i}$, which we can identify with $\mathcal{C}$;

-   $1$-morphisms in ${\sc\mbox{S}\hspace{1.0pt}}$ are all endofunctors of $\mathcal{C}$ which are isomorphic to endofunctors given by tensoring with Soergel $\mathtt{C}\text{-}\mathtt{C}$–bimodules;

-   $2$-morphisms in ${\sc\mbox{S}\hspace{1.0pt}}$ are natural transformations of functors (these correspond to homomorphisms of Soergel $\mathtt{C}\text{-}\mathtt{C}$–bimodules).

The $2$-category ${\sc\mbox{S}\hspace{1.0pt}}$ is fiat. The algebra $A_{{\scc\mbox{S}\hspace{1.0pt}}}$ is isomorphic to $\mathbb{C}[W]$ and is positively based with respect to the Kazhdan-Lusztig basis in $\mathbb{C}[W]$. For $w\in W$, let $\theta_w$ denote a fixed representative in the isomorphism class of indecomposable $1$-morphisms in ${\sc\mbox{S}\hspace{1.0pt}}$ given by tensoring with $B_w$.

Small quotients of Soergel bimodules {#s3.2}
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Let $(W,S)$, $V$, $\mathcal{C}$ and ${\sc\mbox{S}\hspace{1.0pt}}$ be as above.

\[lem1\] For any $s,t\in S$, we have $\theta_s\sim_J\theta_t$.

As $(W,S)$ is irreducible, we may assume that $st\neq ts$. As $\theta_s\theta_t\cong \theta_{st}$, we have $\theta_s\leq_R\theta_{st}$. Further, as $st\neq ts$, we have $\theta_{st}\theta_s\cong \theta_{sts}\oplus \theta_s$. This implies that 