Every monoidal category is strictly monoidally isomorphic to a strict one

Suppose a strict monoidal isomorphism $F:\mathbf{S}\mathbf{e}\mathbf{t}\to \mathcal{D}$ existed with $\mathcal{D}$ strict. Strict preservation and strict associativity in $\mathcal{D}$ would give $F((A\times B)\times C)=(F(A)\otimes F(B))\otimes F(C)=F(A)\otimes (F(B)\otimes F(C))=F(A\times (B\times C))\mathrm{.}$ Since $F$ is injective on objects, this would force $(A\times B)\times C=A\times (B\times C)$ as literal sets for all $A,B,C$.

For nonempty sets under the usual ordered-pair construction, these two nested cartesian products are not literally equal: their elements have different bracketed pair shapes. This contradicts step 1.1.

Thus cartesian $\mathbf{S}\mathbf{e}\mathbf{t}$ is not strictly monoidally isomorphic to a strict monoidal category, even though [L1] supplies a monoidal equivalence to one. Therefore the claim is false.