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A bialgebra equipped with a coquasitriangular structure, namely a convolution-invertible scalar form $r:H\otimes H\to k$ satisfying $r(ab,c)=\sum r(a,c_{(1)})r(b,c_{(2)})$, $r(a,bc)=\sum r(a_{(1)},c)r(a_{(2)},b)$ and $\sum r(a_{(1)},b_{(1)})a_{(2)}{b}_{(2)}=\sum b_{(1)}{a}_{(1)}r(a_{(2)},b_{(2)})$, with unit normalization. The induced right-comodule braiding is $v\otimes w\mapsto \sum w_{(0)}\otimes v_{(0)}r(v_{(1)},w_{(1)})$.

Given an invertible braiding $C$ on $V\otimes V$, the universal bialgebra coacting on $V$ for which $C$ is a comodule map is generated by the matrix coefficients $t_{ij}$. Its comultiplication is $\Delta t_{ij}={\sum }_k{t}_{ik}\otimes t_{kj}$. Colinearity gives quadratic relations. Writing $R=\tau C$ turns the relation into the familiar RTT construction with the appropriate ordered tensor factors. A Yang-Baxter R-matrix supplies a scalar coquasitriangular structure.

A cobraided bialgebra is a bialgebra $H$ equipped with a convolution-invertible bilinear form $r:H\otimes H\to k$ satisfying the following identities, in Sweedler notation:Convolution invertibility means there is $\bar{r}$ with $\sum r(a_{(1)},b_{(1)})\bar{r}(a_{(2)},b_{(2)})=\epsilon (a)\epsilon (b)$, and likewise with $r,\bar{r}$ interchanged. This is a coquasitriangular structure. On right comodules it induces the braidingThe third identity makes this a comodule map, the first two give its two hexagon identities, and the convolution inverse gives the inverse braiding.

With the right-comodule braiding convention in part (a), the defining formula isConsequently the scalar form on generators is the following table, with row the first argument and column the second:For example, the extra $e_2\otimes e_1$ component of $C(e_2\otimes e_1)$ is $r(b,c)=q^{-1/2}(q-q^{-1})$. The other off-diagonal component gives $r(d,a)=q^{-1/2}$.

For arbitrary elements of the FRT bialgebra, extend this table by bilinearity, the unit rule $r(1,z)=r(z,1)=\epsilon (z)$, andThese are an explicit recursive algorithm for any two words in the generators, so the table determines the entire cobraiding. The braid equation for $C$ ensures the algorithm annihilates the defining coefficient relations in either argument: moving one crossing past two successive crossings gives precisely the two reductions of a quadratic colinearity relation. Thus it descends from the free algebra over a field to the FRT bialgebra. For the convolution inverse, use the inverse of $R=\tau C$, where $\tau$ flips the tensor factors: its coefficients are $\bar{r}(t_{ki},t_{lj})=(R^{-1}{)}_{kl,ij}$. Equivalently, in the flipped coefficient formula use $\tau C^{-1}\tau$, not $C^{-1}$ alone. This gives $\bar{r}(a,a)=\bar{r}(d,d)=q^{-1/2}$, $\bar{r}(a,d)=\bar{r}(d,a)=q^{1/2}$, $\bar{r}(b,c)=-q^{1/2}(q-q^{-1})$, and all other generator values zero. This specifies the coquasitriangular structure completely, not just its values on degree-one elements.

We use right comodules and write their coactions as $u\mapsto \sum u_{(0)}\otimes u_{(1)}$. A coquasitriangular structure gives the braiding on these comodulesIn an arbitrary symmetric monoidal category this notation abbreviates a composite of the two coactions, the ambient symmetry, and $\gamma$; it does not assume that the objects have elements. The coquasitriangular axioms ensure that this composite is a comodule morphism, is invertible using the convolution inverse of $\gamma$, and satisfies the two hexagon laws. More explicitly, in scalar notation those laws come fromwhile the comodule-morphism condition isUnit normalizations give the unit constraints. These descriptions are identities of morphisms in the ambient symmetric category, with the displayed reordering carried by its symmetry.

Apply the preceding self-braiding result to the regular right comodule $(H,\Delta )$. Its Yang–Baxter operator isCompose its braid equation with the three counits. Expanding $R_{23}{R}_{12}{R}_{23}$ and cancelling the leading counit factors givesExpanding $R_{12}{R}_{23}{R}_{12}$ instead givesThey are equal by the Yang–Baxter operator equation. This is the required scalar identity:Indeed, after ${\Delta }^{\otimes 3}$ the ordered factors are $a_{(1)},a_{(2)},b_{(1)},b_{(2)},c_{(1)},c_{(2)}$. The two ambient symmetries on the left produce the three pairings $(a_{(1)},b_{(1)})$, $(a_{(2)},c_{(1)})$, $(b_{(2)},c_{(2)})$. On the right, the central symmetry produces $a_{(1)},a_{(2)},b_{(2)},b_{(1)},c_{(1)},c_{(2)}$, yielding exactly the other three pairings. Hence the calculation identifies the actual morphisms requested, also when the category is not a category of vector spaces.