Coherence theorem for monoidal categories in string diagram calculus

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In this paper Theorem 3.1 is a coherence theorem for monoidal categories in the string diagram calculus.

I don't mind taking this theorem for granted for now, but I'm not quite sure whether I really understand it: the statement sounds too easy, so much that I'm quite unsure if I'd be using it right.

The Caveat 3.2. following the theorem confuses me even more.

Where can I find a more comprehensive / rigorous explanation of the coherence theorem for monoidal categories using string diagrams?