Anindya, John

Thanks for clarification. I was at first intimidated by the proof so didn't dive in but after your reading your comments and sketches, I went through it and can see what is going on. Basically, the theorem proved that \$$\mathcal{C}\$$ and \$$\mathrm{str}(\mathcal{C})\$$ are equivalent by showing there are two functors that are inverses between them. Hence, we can go back and forth between \$$\mathcal{C}\$$ and \$$\mathrm{str}(\mathcal{C})\$$ to move parentheses around. Indeed, all of the original isomorphisms are still there and the strict monoidal category also has isomorphisms within them so cannot be skeletal.