Let's say we have a duality between two objects $X$ and $Y$ in a braided monoidal category given by $c \colon \mathbb 1 \to X \otimes Y$, $e \colon Y \otimes X \to \mathbb 1$. I was wondering whether $\mathbb 1 \xrightarrow{c} X \otimes Y \xrightarrow{B_{X, Y}} Y \otimes X$ and $X \otimes Y \xrightarrow{B_{X, Y}} Y \otimes X \xrightarrow{e} \mathbb 1$ (where $B$ is the braiding) also give rise to a duality.
The nlab page on dualizable objects suggests that notions of left and right dualizability agree in braided monoidal categories:
If is braided then left and right adjoints in are equivalent; otherwise one speaks of being left dualizable or right dualizable.
So I thought that this might work out and started a diagram chase, but ended up needing to show that $$Y \otimes (X \otimes Y) \xrightarrow{B_{Y, X \otimes Y}} (X \otimes Y) \otimes Y \cong X \otimes (Y \otimes Y) \xrightarrow{B_{X, Y \otimes Y}} (Y \otimes Y) \otimes X \cong Y \otimes (Y \otimes X) \xrightarrow{B_{Y, Y \otimes X}} (Y \otimes X) \otimes Y \cong Y \otimes (X \otimes Y)$$ is the identity, which (I think) doesn't necessarily hold if the braiding doesn't "square to the identity".
Do we really need a symmetric monoidal category for this "swapped" duality to work or is there another way of exhibiting $X$ and $Y$ as duals of each other "in the other direction"?