\Dg v_1\xc \otimes v_k & \rMapsto & v_k\xc \otimes v_1 \\ T(V) & \rTo & T(V) \\ \dTo <\theta & & \dTo >\theta \\ C(V,q) & \rTo & C(V,q) \\ v_1\cdots v_k & \rMapsto & v_k\cdots v_1 & {} \equiv (v_1\cdots v_k)^* \\ \endDg