The type-A diagrammatic and bimodule Soergel categories are equivalent

Hypotheses and the functor: by [F5] the type-A realization over $\mathbb{Q}$ with $n\ge 2$ satisfies exactly the standing hypotheses of the imported results, so the functor statement [F1] and the two basis statements [F2] and [F3] apply in the stated form; by [F1] $\mathcal{F}$ is a graded $\mathbb{Q}$-linear monoidal word functor to total graded Hom with $\underset{‾}{i}\mapsto B_{\underset{‾}{i}}$ and $\mathrm{\varnothing }\mapsto R$. Its finite-sum and shifted extension restricts to degree-zero morphisms before idempotent completion, yielding $\mathrm{Kar}(\mathcal{F}):\mathrm{Kar}(D)\to \mathrm{Kar}(\mathrm{B}\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m})={\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$, given on objects by $(M,e)\mapsto (\mathcal{F}M,\mathcal{F}e)$ and on morphisms by $u\mapsto \mathcal{F}u$; the degree-zero idempotent $\mathcal{F}e$ is well defined because $\mathcal{F}$ preserves degrees, composition and identities.

Bijection on the homs of $D$: fix words $\underset{‾}{x},\underset{‾}{y}$; by [F2] the double leaves are a homogeneous free left $R$-basis of ${\mathrm{Hom}}_D(\underset{‾}{x},\underset{‾}{y})$ and by [F3] their images under $\mathcal{F}$ are a homogeneous free left $R$-basis of ${\mathrm{Hom}}_{R\text{-}R}(B_{\underset{‾}{x}},B_{\underset{‾}{y}})$ with the same indexing and the same degrees, so the $\mathbb{Q}$-linear map $g\mapsto \mathcal{F}(g)$ takes a basis to a basis of free graded $R$-modules of the same graded rank and is therefore an isomorphism of graded left $R$-modules. For finite sums of shifted words, Hom maps are matrices of shifted word-Hom entries, and [F3] gives the same matrix basis after evaluation; thus the isomorphism extends to total graded Hom for these sums and restricts to a bijection on its degree-zero part.

Monoidal and graded structure: by [F1] $\mathcal{F}$ is a graded monoidal functor, so $\mathcal{F}(X\otimes Y)\cong \mathcal{F}X\otimes \mathcal{F}Y$ compatibly with the associativity and unit constraints, $\mathcal{F}(\mathrm{\varnothing })=R$ is the unit, and $\mathcal{F}(X\{k\})=(\mathcal{F}X)\{k\}$; the tensor product and shifts on the idempotent completions are $(M,e)\otimes (N,f)=(M\otimes N,e\otimes f)$ and $(M,e)\{k\}=(M\{k\},e\{k\})$ by [F4], and $\mathcal{F}(e\otimes f)=\mathcal{F}e\otimes \mathcal{F}f$ because $\mathcal{F}$ is monoidal; hence $\mathrm{Kar}(\mathcal{F})$ is monoidal and preserves the shifts, and it preserves degrees of morphisms by [F1].

Faithful on the Karoubi envelope: let $(M,e)$ and $(N,f)$ be objects of $\mathrm{Kar}(D)$ and let $u:(M,e)\to (N,f)$ be a degree-zero morphism, so that $fu=u=ue$ by [F6]; applying $\mathcal{F}$ gives $\mathcal{F}(f)\mathcal{F}(u)=\mathcal{F}(u)=\mathcal{F}(u)\mathcal{F}(e)$, so $\mathcal{F}u$ is a categorical morphism $(\mathcal{F}M,\mathcal{F}e)\to (\mathcal{F}N,\mathcal{F}f)$ and $\mathrm{Kar}(\mathcal{F})$ on this hom set is the restriction of the degree-zero bijection of step 1.2 to these subgroups; a restriction of an injective map is injective, so $\mathrm{Kar}(\mathcal{F})$ is faithful.

Full on the Karoubi envelope: let $v:(\mathcal{F}M,\mathcal{F}e)\to (\mathcal{F}N,\mathcal{F}f)$ be a degree-zero morphism of ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$, so that $v=\mathcal{F}(f)v\mathcal{F}(e)$ by [F4]; by the degree-zero surjectivity in step 1.2 there is a degree-zero $u\in {\mathrm{Hom}}_D(M,N)$ with $\mathcal{F}(u)=v$, and then $\mathcal{F}(fue)=\mathcal{F}(f)\mathcal{F}(u)\mathcal{F}(e)=v=\mathcal{F}(u)$; since $\mathcal{F}$ is injective on degree-zero ${\mathrm{Hom}}_D(M,N)$ by step 1.2, $fue=u$, so $u$ is a morphism $(M,e)\to (N,f)$ of $\mathrm{Kar}(D)$ with $\mathrm{Kar}(\mathcal{F})(u)=v$. Hence $\mathrm{Kar}(\mathcal{F})$ is full.

Essential surjectivity: let $(M,e)$ be an object of ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$; by [F4] $M$ is a finite direct sum of shifts of Bott–Samelson products, so $M=\mathcal{F}(X)$ for the corresponding finite direct sum $X$ of shifts of words in the additive closure of $D$, by the additivity and shift preservation of the extension in [F1]; the degree-zero idempotent $e\in \mathrm{End}(M)$ has a degree-zero preimage $\tilde{e}\in {\mathrm{End}}_D(X)$ under the bijection of step 1.2, and $\tilde{e}$ is idempotent because $\mathcal{F}({\tilde{e}}^2)=\mathcal{F}(\tilde{e}{)}^2=e^2=e=\mathcal{F}(\tilde{e})$ and $\mathcal{F}$ is faithful by step 2.1; therefore $(M,e)=(\mathcal{F}X,\mathcal{F}\tilde{e})=\mathrm{Kar}(\mathcal{F})(X,\tilde{e})$ and $\mathrm{Kar}(\mathcal{F})$ is essentially surjective.

Conclusion: steps 2.1, 3.1, 3.2 and 1.3 show that $\mathrm{Kar}(\mathcal{F})$ is faithful, full, essentially surjective and a graded monoidal functor, hence an equivalence of graded monoidal categories; the realization hypotheses used are exactly those recalled in [F5], and for $n\le 1$ both sides are generated under finite sums, shifts and summands by $R$, with ${\mathrm{End}}^{\bullet }(R)=R$ and categorical ${\mathrm{End}}^0(R)=\mathbb{Q}$, and $\mathcal{F}$ is the identity on this generator, so the conclusion holds in that case too. ∎