This won't fit in a comment, maybe it's useful to someone:

$$ \begin{align*}
(\mathrm{id}_Y⊗\mathrm{ev}_Y)∘(\mathrm{coev}_Y⊗\mathrm{id}_Y) &= 
(\mathrm{id}_Y⊗\mathrm{ev}_X)∘(\mathrm{id}_{Y⊗X^*}⊗σ^{-1})
∘(σ⊗\mathrm{id}_{X^*⊗Y})∘(\mathrm{coev}_X⊗\mathrm{id}_Y) \\
&= (\mathrm{id}_Y⊗\mathrm{ev}_X)∘(σ⊗σ^{-1})∘(\mathrm{coev}_X⊗\mathrm{id}_Y) \\
&= (σ⊗\mathrm{id}_1)∘(\mathrm{id}_X⊗\mathrm{ev}_X)∘(\mathrm{coev}_X⊗\mathrm{id}_X)∘(\mathrm{id}_1⊗σ^{-1}) \\
&= σ∘σ^{-1}
\end{align*}$$

The point is that the unitors are usually suppressed in the triangle identities.