There are two common variations of definition 2.1.1: 
Namely, \(\mathrm{BSbim}_n\) is the \(k\)-linear (but no longer graded \(k\)-linear) category obtained by restricting to the degree zero part of the morphism spaces. We call this the degree zero subcategory. (In the language of enriched category theory, this is the underlying category of the category enriched in graded vector spaces; it inherits the linear structure.) By remembering the \(\mathbb{Z}\)-action by grading shift functors, all other homogeneous components of morphism spaces in \(\mathrm{BSbim}^{\mathrm{gr}}_n\) can be recovered from ([000K]).
Alternatively, one can consider the graded \(k\)-linear full subcategory \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\) on unshifted Bott–Samelson bimodules \(B_{\mathbf{i}}\), i.e. where \(j=0\) in ([000M]). From this category one can reconstruct the morphism spaces between shifted Bott–Samelsons, that is all objects in \(\mathrm{BSbim}^{\mathrm{gr}}_n\), again via ([000K]).
We refer to [MOS09, (2.1)] for a discussion of these essentially equivalent ways of handling graded \(k\)-linear categories.