Coherent graded biadjunction and the crossing mates

Explicit units, counits and triangles. Put $A=R^{\mathrm{\Lambda }}(\beta )$, $B=R^{\mathrm{\Lambda }}(\beta +{\alpha }_i)$, $e=e(\beta ,i)$, $X=Be$, $Y=eB$ and $t=t_i(\beta )$. The ordinary first unit is the append homomorphism ${\iota }_i:A\to eBe$, $a\mapsto {\iota }_i(a)$, and its counit is multiplication; the map need not be injective. On a module the unit is $m\mapsto e\otimes m$, and both triangles follow from tensor balancing and multiplication, including when $e=0$. For the second adjunction use the actual map $y{\otimes }_{B}x\mapsto T_i(yx)$ and unit $b\mapsto b{C}_i$ of [F1], tensoring with the input module. The balancing uses $A$-bilinearity of $T_i$ and $B$-centrality of $C_i$. Its two triangle composites send $x$ to $\sum x_{\alpha }{T}_i(y_{\alpha }x)=x$ and $y$ to $\sum T_i(y{x}_{\alpha })y_{\alpha }=y$. Their degrees are respectively $t$ and $-t$, so the resulting degree-zero adjunction is $E_i\{t\}⊣F_i$, together with $F_i⊣E_i$. These exact bimodule identities hold after tensoring with every graded module; no finite-projective-only extension or quotient flatness is assumed.

Actual crossing compatibility. The cap/Casimir maps in step 1.1 were transported through the canonical first-adjoint comparison, so [F1] gives native dot and equal-color mate cyclicity. For unequal colors, let $p_{\nu }={\tau }_{r}e(\nu )\in e(s_r\nu )Be(\nu )$ with bottom pair $(i,j)$ and $a\in e(\nu )Be(s_r\nu )$. The common residual in [F2] evaluates $a{p}_{\nu }$ in the $ij$ corner with scalar $t_{ji}$ and $p_{\nu }a$ in the $ji$ corner with scalar $t_{ij}$. Close a suffix first using actual cap bilinearity and then the unchanged prefix to obtain $\mathrm{cl}(p_{\nu }a)=(t_{ij}\mathrm{/}{t}_{ji})\mathrm{cl}(a{p}_{\nu })$. The constructed perfect opposite-corner word pairing and unit/counit evaluation compatibility identify the double right mate with $(t_{ij}\mathrm{/}{t}_{ji})p_{\nu }$. This fixes its direction under bottom-to-top operator composition; reversing the chosen crossing exchanges the ratio. Passing from source $\kappa$ to native $\tau =-\kappa$ negates both paired single crossings and does not change the ratio. No conclusion is inferred from assigning different scalars to the two orders of one fixed partial pair. Apply the actual mate equality to every vacuum word projective of the input algebra. Their finite sum is its regular module, so the equality is one of tensor-bimodule maps, and tensoring gives it on arbitrary modules.

Boundaries, monic quotients and conclusion. The construction of [F1] includes zero append corners, where both functors and tensors are zero, and the vacuum monic coefficient cap and dual-power Casimir. All maps and both triangles are integral monic coefficient identities, and the full terminal comparison of [F2] descends to the ordinary polynomial algebra before quotienting. Arbitrary monic field specialization therefore retains the same maps, triangles and ordered mates with shifts forgotten. In the standard homogeneous branch the cap/Casimir degrees in step 1.1 give the displayed $t_i(\beta )$ at every weight. Thus the full stated biadjunction and native ordered mate identities hold on all graded modules, with no finite-type, characteristic-zero, Frobenius, Grothendieck-module or crystal premise.