The intertwiner induced by a braided monoidal functor

Let $F\text{\hspace{0.17em}⁣}:\mathcal{C}\to \mathcal{D}$ be a braided monoidal functor between braided monoidal categories (Braided monoidal functor), with binary tensor constraint $J_{X,Y}\text{\hspace{0.17em}⁣}:F(X)\otimes F(Y)\to F(X\otimes Y)$ and unit constraint $J_0\text{\hspace{0.17em}⁣}:\mathbf{1}\to F(\mathbf{1})$, which are isomorphisms because $F$ is strong monoidal (Lax, strong, and strict monoidal functors). Let $X\in \mathcal{C}$.

For $n\ge 1$ the $n$-fold constraint is the canonical isomorphism

$J_n\text{\hspace{0.17em}⁣}:F(X{)}^{\otimes n}⟶F(X^{\otimes n}),$

defined in a strict model of both categories by the recursion $J_1=1_{F(X)}$ and $J_{n+1}=J_{X^{\otimes n},X}\circ (J_n\otimes 1_{F(X)})$; in general the associativity and unit isomorphisms of the two monoidal structures are inserted in the same composite, and $J_n$ is independent of those insertions by Mac Lane coherence for the monoidal structure.

We call $J_n$ the intertwiner induced by $F$ at $X$. It conjugates the canonical braid action on $F(X{)}^{\otimes n}$ (An object of a braided category carries canonical braid actions) to $F$ of the canonical braid action on $X^{\otimes n}$: the precise statement is that $J_n\circ {\rho }_n^{\mathcal{D}}(\beta )=F({\rho }_n^{\mathcal{C}}(\beta ))\circ J_n$ for every $\beta \in B_n$, which is proved in the companion theorem. Each $J_n$ is an isomorphism because it is a composite of the structure isomorphisms $J_{X,Y}$ and $J_0$, which are invertible by strong monoidality; in particular $J_n$ is a natural isomorphism between the two tensor-power functors.