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Here the unqualified term allows lax comparison maps $F_2:FX\otimes FY\to F(X\otimes Y)$ and $F_0:I\to FI$, natural and compatible with associators and unitors. Invertible comparison maps give a strong monoidal functor. The opposite direction gives an opmonoidal functor.

Define the length of a formal tensor expression recursively byThe canonical strict monoidal functor $Q:\mathcal{F}br\to \mathbf{B}$ sends $u$ to $|u|$, sends every associator and unitor to an identity braid, and sends $c_{u,v}$ to the block braiding $c_{|u|,|v|}$. The pentagon and triangle become identity equations; the braiding axioms become the corresponding block-braid equations. Thus the assignment respects the defining relations, andon the nose. It is a braided monoidal functor with identity comparison maps.

To prove that $Q$ is an equivalence of categories, choose a standard parenthesization $x^{[n]}$ of $n$ copies of $x$, with $x^{[0]}=I$. On $x^{[n]}$, define the image of ${\sigma }_i$ by canonically exposing the $i$th and $(i+1)$st factors, applying $c_{x,x}$ there, and restoring the chosen parentheses. The monoidal coherence theorem makes this independent of the structural rebracketing. Crossings on disjoint pairs commute by the tensor interchange law. The adjacent braid group relations follows from the hexagon laws and naturality of the braiding, as in the Yang–Baxter calculation below. Hence these assignments give group homomorphisms $B_n\to \mathrm{Aut}(x^{[n]})$ and a functor $S:\mathbf{B}\to \mathcal{F}br$.

Equip $S$ with the canonical rebracketing maps $x^{[m]}\otimes x^{[n]}\to x^{[m+n]}$. Their monoidal functor axioms follow from the monoidal coherence theorem; the block-braiding compatibility follows by iterating the two hexagon laws. Therefore $S$ is a strong monoidal functor compatible with the braiding.

We have $QS=1_{\mathbf{B}}$, including its comparison maps. For every formal expression $u$, there is a canonical structural isomorphismobtained by rebracketing and inserting the units appearing in $u$. These maps form a natural isomorphism $SQ\Rightarrow 1_{\mathcal{F}br}$. To check naturality, it suffices to check the generating morphisms: for associators and unitors it is exactly monoidal coherence; for $c_{u,v}$ it follows from the hexagon expansion into the elementary crossings defining $S$. Composition and tensor product then preserve the equation. The same structural coherence shows that $\theta$ is monoidal.

Thus $Q$ has a specified quasi-inverse and is the required equivalence:Only ordinary monoidal coherence, together with the defining braiding axioms, has been used; a separate braided coherence theorem is unnecessary.

Because the direction of a lax monoidal functor compares the target tensor with the image of the source tensor, the given maps ${\phi }_{X,Y}:X\bullet Y\to X\diamond Y$ have the correct direction. They need not be invertible unless a strong monoidal functor is intended.

Their associativity condition is the following commutative diagram, with the associators of the two structures labelled explicitly:The two unit conditions, since ${\phi }_0=1_I$, are the unit commutative diagramsTogether with the assumed naturality, these are exactly the axioms for the requested monoidal structure. There are no further coherence conditions.

The faithful strict monoidal functor $U$ allows all three diagrams to be checked after applying it. Both tensor structures then have the same underlying tensor and constraints in $\mathcal{V}$. In particular, the unit conditions are equivalent toThe associativity square becomes the displayed diagram with every tensor replaced by $\otimes$, both associators by ${\alpha }^{\mathcal{V}}$, and every $\phi$ by $U\phi$. Faithfulness reflects equality of the two resulting composites. One should not identify ${\phi }_{I,X}$ itself with an identity in $\mathcal{C}$: its domain and codomain can be distinct objects with the same image under $U$. Intrinsically it is $({\lambda }_X^{\diamond }{)}^{-1}{\lambda }_X^{\bullet }$, and similarly on the right.

A strong monoidal functor whose comparison maps are identities. It preserves the tensor, unit and structural constraints exactly; its source need not be a strict monoidal category.