Derived comparisons give unique normalized homotopy maps

Given: Signed words $t,u$ with product $w$, the word complexes $F(t),F(u)$ of The Rouquier complex of a braid word, and the generator relations of Opposite Rouquier generator complexes are homotopy inverse, Rouquier complexes satisfy far commutativity, Rouquier complexes satisfy the three-term braid relation.

Invertibility of word complexes. Every word complex $F(t)$ is invertible in $K^b(R^e\text{-grmod})$: $F(t){\otimes }_{R}F(t^{-1})\simeq R\simeq F(t^{-1}){\otimes }_{R}F(t)$, where $t^{-1}$ is the reversed word with inverted signs; this follows from the generator relations by induction on the length of the word, tensoring the identities $F_i{\otimes }_R{F}_i^{-1}\simeq R$, $F_i{\otimes }_R{F}_j\cong F_j{\otimes }_R{F}_i$ for distant $i,j$ and the three-term relation. By the Artin presentation The braid group by Artin presentation, equal braid words differ by finitely many relation replacements and inverse-pair insertions or deletions: their quotient in the free group is a finite product of conjugates of relators. Tensoring the generator equivalences in those word contexts therefore compares any two words for the same braid. The same relations hold after localization, with tensoring by these two-sided finite-free complexes computed by ordinary totalization. (Opposite Rouquier generator complexes are homotopy inverse, Rouquier complexes satisfy far commutativity, Rouquier complexes satisfy the three-term braid relation, The Rouquier complex of a braid word)

The unit and its endomorphisms. A degree-zero bimodule map $R\to R$ is multiplication by its value at $1$, which must lie in $R_0=\mathbb{Q}$. The unit complexes have no possible nonzero chain homotopies, so ${\mathrm{Hom}}_{K^b}(R,R)=\mathbb{Q}\cdot \mathrm{i}\mathrm{d}$. Since both objects are modules in degree zero, their degree-zero derived-category Hom is the ordinary module Hom, giving ${\mathrm{Hom}}_{D^b}(R,R)=\mathbb{Q}\cdot \mathrm{i}\mathrm{d}$ as well: apply the boundary Hom formula of The canonical pair is a t structure with $a=b=0$ to graded $R^e$-modules, so both cohomological and internal degrees are zero (The homotopy category of chain complexes, Derived category of an abelian category, Standard graph bimodules, support filtrations and characters). For $Z\simeq R$, an isomorphism in the homotopy category transports ${\mathrm{Hom}}_{K^b}(R,Z)$ to ${\mathrm{Hom}}_{K^b}(R,R)$; it does not assert a generic identification with all of $H^0(Z{)}_0$.

Tensoring with an invertible object. Let $\mathcal{C}$ be a monoidal category and let $X\in \mathcal{C}$ admit a two-sided inverse $X^{-1}$: isomorphisms $X\otimes X^{-1}\to \mathbf{1}$ and $X^{-1}\otimes X\to \mathbf{1}$. Using the associativity and unit isomorphisms of $\mathcal{C}$, these exhibit natural isomorphisms $(-\otimes X^{-1})(-\otimes X)\cong {\mathrm{i}\mathrm{d}}_{\mathcal{C}}$ and $(-\otimes X)(-\otimes X^{-1})\cong {\mathrm{i}\mathrm{d}}_{\mathcal{C}}$; hence $-\otimes X$ is an equivalence of categories with quasi-inverse $-\otimes X^{-1}$ in the sense of Equivalence, quasi-inverse, and adjoint equivalence of categories. By Every equivalence of categories can be equipped as an adjoint equivalence the pair can be equipped as an adjoint equivalence, and the adjunction then gives, by An adjoint equivalence is an adjunction whose unit and counit are natural isomorphisms together with The unit-counit, hom-set, unit-universal, and counit-universal encodings of an adjunction are equivalent, a natural bijection $\mathrm{Hom}(U\otimes X,W)\cong \mathrm{Hom}(U,W\otimes X^{-1})\mathrm{.}$ In $K^b(R^e\text{-grmod})$ and $D^b(R^e\text{-grmod})$ the associativity and unit isomorphisms are those of Bounded bimodule tensor is associative, unital, and compatible with cones and their images under localization, so the bijection is available in both categories.

Derived graph models. $F(t)\cong R_{\pi (w)}(-e(t))$ and $F(u)\cong R_{\pi (w)}(-e(u))$ in $D^b(R^e\text{-grmod})$, and the comparison $c_{t,u}$ of the two words induces an isomorphism of these models whose class in ${\mathrm{Hom}}_{D^b}$ is nonzero; since the two Artin relations have equal exponent sums on both sides and inverse pairs have exponent zero, the exponent is invariant on braid words and $e(t)=e(u)$ and the two graph models coincide. (Rouquier generator complexes have canonical derived graph models, Canonical comparisons between standard graph tensor products)