TITLE: What is the proof of the compatibility of a braiding with the unitors?
QUESTION [5 upvotes]: I am specifically referencing the property that, given a braided monoidal category with a braiding $c$ and left and right unitors $\lambda, \rho$,
$$
\lambda_A \circ c_{A,I}=\rho_{A},
$$
for any object $A$. This equation is stated in almost every reference (first done so by Joyal and Street) defining a braided category, yet an explicit or reproducible proof is, to my knowledge, not given anywhere. The closest thing to a proof I can find is a series of hints on Mathematics SE asked in 2014, however this answer (by Turion) has several inconsistencies/typos, making it hard to understand. Is there some reference where this proof is actually 'spelled out'? How can one prove this relation given the braided monoidal category axioms?

REPLY [5 votes]: A completely spelled out diagrammatic proof is given in Proposition 1.3.21 of Volume II of Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory by Niles Johnson and Donald Yau.