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Your proposed theorem is totally false (somehow, I think the half-bakedness is not in Turaev's book). The ribbon structure is used, since you use both the maps $b$ and $d$; a duality on your category only gives you one of these.

EDIT: I misread the notation Turaev's book. What I actually meant was that you need to the ribbon structure to be able to attach maps to the left-oriented cups and caps, denoted $\cap^-$ and $\cup^-$.

In $\mathrm{Rib}_{\mathcal{V}}$ we have that $(V,+) \otimes (W,+)\cong (W,+) \otimes (V,+)$ via the crossing; thus any small monoidal category with duals that contains objects $V$ and $W$ such that $V\otimes W\ncong W\otimes V$ gives a counter-example. If you want a particular one, take the strictification of f.d. representations of a KLR algebra.

EDIT: I slightly misread the statement of the proposed theorem above (I didn't notice the switch in order from Turaev's statement), but it's still false. Let $F$ be the identity functor. Then $\overline{F}$ must be a functor sending $(V,+) \mapsto V$, so it sends the crossing to an isomorphism $V\otimes W\cong W\otimes V$.

EDIT: I misunderstood what the OP was suggesting. This is probably true. I doubt anyone has bothered to write it, though I could be wrong. I suspect it's "well-known." The isotopies should just come from the Reidemeister moves, and sliding coupons across strands, which all are indeed known to be commutative diagrams in all ribbon categories.

Your proposed theorem is totally false (somehow, I think the half-bakedness is not in Turaev's book). The ribbon structure is used, since you use both the maps $b$ and $d$; a duality on your category only gives you one of these. In $\mathrm{Rib}_{\mathcal{V}}$ we have that $(V,+) \otimes (W,+)\cong (W,+) \otimes (V,+)$ via the crossing; thus any small monoidal category with duals that contains objects $V$ and $W$ such that $V\otimes W\ncong W\otimes V$ gives a counter-example. If you want a particular one, take the strictification of f.d. representations of a KLR algebra.

EDIT: I slightly misread the statement of the proposed theorem, but it's still false. Let $F$ be the identity functor. Then $\overline{F}$ must be a functor sending $(V,+) \mapsto V$, so it sends the crossing to an isomorphism $V\otimes W\cong W\otimes V$.

Your proposed theorem is totally false (somehow, I think the half-bakedness is not in Turaev's book). The ribbon structure is used, since you use both the maps $b$ and $d$; a duality on your category only gives you one of these.

EDIT: I misread the notation Turaev's book. What I actually meant was that you need to the ribbon structure to be able to attach maps to the left-oriented cups and caps, denoted $\cap^-$ and $\cup^-$.

In $\mathrm{Rib}_{\mathcal{V}}$ we have that $(V,+) \otimes (W,+)\cong (W,+) \otimes (V,+)$ via the crossing; thus any small monoidal category with duals that contains objects $V$ and $W$ such that $V\otimes W\ncong W\otimes V$ gives a counter-example. If you want a particular one, take the strictification of f.d. representations of a KLR algebra.

EDIT: I slightly misread the statement of the proposed theorem above (I didn't notice the switch in order from Turaev's statement), but it's still false. Let $F$ be the identity functor. Then $\overline{F}$ must be a functor sending $(V,+) \mapsto V$, so it sends the crossing to an isomorphism $V\otimes W\cong W\otimes V$.

Your proposed theorem is totally false (somehow, I think the half-bakedness is not in Turaev's book). The ribbon structure is used, since you use both the maps $b$ and $d$; a duality on your category only gives you one of these. In $\mathrm{Rib}_{\mathcal{V}}$ we have that $(V,+) \otimes (W,+)\cong (W,+) \otimes (V,+)$ via the crossing; thus any small monoidal category with duals that contains objects $V$ and $W$ such that $V\otimes W\ncong W\otimes V$ gives a counter-example. If you want a particular one, take the strictification of f.d. representations of a KLR algebra.

Your proposed theorem is totally false (somehow, I think the half-bakedness is not in Turaev's book). The ribbon structure is used, since you use both the maps $b$ and $d$; a duality on your category only gives you one of these. In $\mathrm{Rib}_{\mathcal{V}}$ we have that $(V,+) \otimes (W,+)\cong (W,+) \otimes (V,+)$ via the crossing; thus any small monoidal category with duals that contains objects $V$ and $W$ such that $V\otimes W\ncong W\otimes V$ gives a counter-example. If you want a particular one, take the strictification of f.d. representations of a KLR algebra.

EDIT: I slightly misread the statement of the proposed theorem, but it's still false. Let $F$ be the identity functor. Then $\overline{F}$ must be a functor sending $(V,+) \mapsto V$, so it sends the crossing to an isomorphism $V\otimes W\cong W\otimes V$.