The ribbon trace equals the framed-closure evaluation

The word of the closed diagram. Read the closed framed diagram ${\widehat{\beta }}^{\mathrm{f}\mathrm{r}}$ as a word in the elementary generators of the framed oriented tangle category: the diagram of $\beta$ contributes its crossings, and the closing bands contribute, at the free ends, one coevaluation and one evaluation pair together with the crossings and twists produced by the blackboard framing of the closing bands. Since $F_X$ is monoidal [L2], its value on the closed diagram is the composite of the corresponding generator values: evaluations $\mathrm{ev}$, coevaluations $\mathrm{coev}$, braidings $c$ and twists $\theta$. By the closure corollary of [F1] this composite is exactly Turaev's trace: $F_X({\widehat{\beta }}^{\mathrm{f}\mathrm{r}})=\mathrm{tr}({\rho }_n(\beta ))={\mathrm{ev}}_{X^{\otimes n}}\circ c_{X^{\otimes n},(X^{\otimes n}{)}^{\vee }}\circ (({\theta }_{X^{\otimes n}}{\rho }_n(\beta ))\otimes 1_{(X^{\otimes n}{)}^{\vee }})\circ {\mathrm{coev}}_{X^{\otimes n}},$ where ${\rho }_n(\beta )=F_X$ applied to the $(n,n)$-tangle of $\beta$ by the generator values of [L2].

The trace formula equals the library trace. By [F2] the composite of [F1] equals ${\mathrm{Tr}}_L({\psi }_{X^{\otimes n}}{\rho }_n(\beta ))$ with $\psi =u\theta$: expanding ${\psi }_X=u_X{\theta }_X$ by the defining composite of the Drinfeld morphism and substituting into [L4], the evaluation–coevaluation pair introduced by $u$ is cancelled against the outer evaluation by the zig-zag identities of [L3], leaving precisely Turaev's composite. Hence $F_X({\widehat{\beta }}^{\mathrm{f}\mathrm{r}})={\mathrm{Tr}}_L(j_{X^{\otimes n}}{\rho }_n(\beta ))=t_n(\beta )$ by [L1].

Invariance and the curl. Since $F_X$ is a functor and isotopic framed tangles are equal morphisms of the framed oriented tangle category [L2], the value $F_X({\widehat{\beta }}^{\mathrm{f}\mathrm{r}})$ depends only on the framed isotopy class of the closure; by step 2.1 the same holds for $t_n(\beta )$. The closure of ${\iota }_n(\beta ){\sigma }_n^{\pm 1}$ is obtained from the closure of $\beta$ by adding one crossing and one band to the last strand, which in the blackboard framing is the insertion of one full twist ${\phi }_X^{\pm 1}$ on a band; by the generator values of [L2] its image is ${\theta }_X^{\pm 1}$, and with the declared convention the positive stabilization corresponds to the positive curl ${\phi }_X^{+}$ with $F_X({\phi }_X^{+})={\theta }_X$.

Conclusion. Steps 1.1 and 2.1 identify the ribbon trace with the functor's value on the blackboard-framed closure, step 3.1 records the framed-isotopy invariance and the local curl picture used by the stabilization lemma. Multiplicativity and cyclicity of the trace, where used, are the published properties of Basic properties of the categorical trace. The only choice principle used is ${\mathrm{A}\mathrm{C}}_{\omega }$, consumed through the existence of the functor $F_X$ of [L2], which rests on the classification input of the tangle lemma; with $F_X$ available the trace computations of steps 1.1--3.1 are finite and use no further choice.