Proof.
Recall from definition 2.3.1 that \(h_1\mathrm{BSbim}\) has objects \(n \in \mathbb{N}_0\) and endo-hom-sets defined as the subset \(h_0 \mathrm{BSbim}_n \subseteq h_0({}_{R_n} \mathrm{grbmod}_{R_n})\) of isomorphism classes of graded bimodules for the graded polynomial algebra \(R_n = k[x_1, \ldots, x_n]\) with \(x_i\) in degree \(2\) on the Bott-Samelson bimodules. The desired functor to \(h_1\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) follows immediately from this description: It sends an object \(n \in \mathbb{N}_0\) of \(h_1\mathrm{BSbim}\) to the graded polynomial algebra \(R_n= k[x_1,\ldots, x_n]\) and is defined on hom-sets as the full inclusion \(h_0 \mathrm{BSbim}_n \hookrightarrow h_0({}_{R_n}\mathrm{grbmod}^{\mathrm{gr-cp}}_{R_n}).\) Since Bott-Samelson bimodules are graded-compact-projective as right (and left) modules by remark 2.1.3, and since the composition and monoidal structure in \(h_1 \mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) are defined by the relative (underived) tensor product and the tensor product \(\otimes_k\), this indeed defines a faithful monoidal functor. ◻