A monoidal category equivalent to a strict one satisfies coherence

Because $F$ is strong monoidal, applying $F$ to any generator listed in [L1] and then using the structure isomorphisms of $F$ again yields a canonical morphism in $\mathcal{D}$ between the corresponding parenthesised tensor words in the images of the objects. Therefore $F$ sends canonical composites in $\mathcal{C}$ to canonical composites in $\mathcal{D}$.

In the strict target $\mathcal{D}$, [L4] makes every canonical morphism between the same source and target the identity of the common tensor object. Hence the images under $F$ of any two canonical morphisms in $\mathcal{C}$ with the same source and target are equal.

By [L2] and [L3], the underlying functor of $F$ is faithful. So equality after applying $F$ implies equality before applying $F$. Therefore any two canonical morphisms in $\mathcal{C}$ with the same source and target are equal.