In his 2009 survey, Selinger ("A survey of graphical languages for monoidal categories") defines the notion of a 'spacial monoidal category', which (in his graphical calculus) allows one to switch a morphism and a line (an object's identity). The diagrammatic representation is given as:
while the equation for the axiom is: for all $h:I\to I$
$$
\rho_A\circ(1_A\,\otimes h)\,\circ\rho^{-1}_A=\lambda_A\circ(h\otimes 1_A)\circ\lambda^{-1}_A,
$$
where $I$, $\rho$, $\lambda$, and $1_A$ are the unit object, left unitor, right unitor, and identity of $A$ respectively. Later in the paper, Selinger gives the usual definition of a braided monoidal category (p. 14–15). He goes on to make the statement that "every braided monoidal category is spacial; this follows from the naturality (in $I$) of $c_{A,I}:A\otimes I\to I\otimes A$," where $c$ is the braiding.
Making note of just the graphical representation of the equation, I realize that the equation $$ \lambda_A\circ c_{A,I}\circ (1_A\otimes h)\circ c^{-1}_{A, I}\circ\lambda^{-1}_A=\lambda_A\circ (h\otimes 1_A)\circ \lambda^{-1}_A $$ also gives the same diagram as the spacial category axiom, and this equation indeed quite trivially follows from the the naturality condition on the braiding $c$. I am, however, unable to prove the spacial category axiom for a general braided category directly from the axioms of a braided monoidal category; I am not sure even how to begin this proof, since the spacial axiom of course does not reference a braiding, while the naturality condition does. How can one show that the axiom holds from just the braided category axioms?
Note: I should note that most sources on this topic also present graphical 'coherence/correctness theorems' (as does Selinger), which allow one to assert that a well-typed equation of morphisms (those in which both sides have the same domain and codomain) in which the LHS and RHS have the same (or even isotopic) diagrammatic representations are in fact true. Using this coherence theorem, we should be able to assert equality between the LHS of Selinger's axiom equation and the LHS of my equation above. Since my equation above directly follows from naturality as stated, Selinger's then should as well. For this reason, I am confident the equation is indeed true, I am just looking for an algebraic proof that does not rely on the soundness/completeness of the graphical calculus.
UPDATE: Comparing the two LHS's above, one can clearly see that the commutativity of the diagrams below gives a full proof.
Joyal and Street, in their 1993 paper "Braided Tensor Categories", 'prove' the commutativity of these diagrams in Proposition 2.1. They state that one may show, say, the first relation by substituting $I$ for both $B$ and $C$ in the first hexagon condition in the definition of a braided category (shown below),
along with 'using the coherence of $\rho, \lambda$ and $\alpha$ and the invertibility of $c_{A,I}$'. I am now unsure how to prove this fact, having tried the substitution but getting stuck at that point. How does one prove these relations?
