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](https://www.sciencedirect.com/science/article/pii/S0001870883710558)) 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](https://math.stackexchange.com/a/920526/982236) 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?



What is the proof of the compatibility of a braiding with the unitors?
Using the notation of [Joyal and Street](https://www.sciencedirect.com/science/article/pii/S0001870883710558) §2, here’s a proof of $\newcommand{\r}{\rho}\newcommand{\l}{\lambda}\newcommand{\x}{\otimes}\newcommand{\comp}{\!\!\cdot\!}\r\_A = \l\_A \comp c\_{A,I}$. Since $\l$ and $c$ are invertible, it suffices to prove $\r\_A \comp (\l\_A \comp c\_{A,I} \x I) = \l\_A \comp c\_{A,I} \comp(\l\_A \comp c\_{A,I} \x I)$. Notation: I will write composition as $\comp$, and it binds tighter than $\x$; and will write all associators just as $\alpha$, to reduce clutter. I’ll black-box details of steps that are purely about symm. mon. cats., not involving the braiding — these are most easily seen using coherence for SMC’s.


$$\begin{align}
 \r\_A \comp (\l\_A \comp c\_{A,I} \x I)
 &= (\l\_A \comp c\_{A,I}) \comp \r\_{A \x I} & \text{naturality of $\r$} \\
 &= \l\_A \comp c\_{A,I} \comp (A \x \r\_I) \comp \alpha & \text{SMC facts}\\
 &= \l\_A \comp (\r\_I \x A) \comp c\_{A,I \x I} \comp \alpha & \text{naturality of $c$}\\
 &= \l\_A \comp (\r\_I \x A) \comp \alpha \comp (I \x c\_{A,I}) \comp \alpha \comp (c\_{A,I} \x I) & \text{axiom (B1)} \\
 &= \l\_A \comp (I \x \l\_A) \comp (I \x c\_{A,I}) \comp \alpha \comp (c\_{A,I} \x I) & \text{SMC facts}\\
 &= \l\_A \comp (I \x \l\_A \comp c\_{A,I}) \comp \alpha \comp (c\_{A,I} \x I) & \text{functoriality of $\x$}\\
 &= (\l\_A \comp c\_{A,I}) \comp \l\_{A \x I} \comp \alpha \comp (c\_{A,I} \x I) & \text{naturality of $\l$}\\
 &= (\lambda\_A \comp c\_{A,I}) \comp (\lambda\_A \x I) \comp (c\_{A,I} \x I) & \text{SMC facts}\\
 &= (\lambda\_A \comp c\_{A,I}) \comp (\lambda\_A \comp c\_{A,I} \x I) & \text{functoriality of $\x$}
\end{align}
$$


I found this proof using string diagrams, writing down (B1) for $A,I,I$ (as Joyal and Street suggest), and then contemplating how to connect that to the desired equation.


