This won't fit in a comment, maybe it's useful to someone:
$$ \begin{align*} (\mathrm{id}_Y⊗\mathrm{ev}_Y)∘(\mathrm{coev}_Y⊗\mathrm{id}_Y) &= (\mathrm{id}_Y⊗\mathrm{ev}_X)∘(\mathrm{id}_{Y⊗X^*}⊗σ^{-1}) ∘(σ⊗\mathrm{id}_{X^*⊗Y})∘(\mathrm{coev}_X⊗\mathrm{id}_Y) \\ &= (\mathrm{id}_Y⊗\mathrm{ev}_X)∘(σ⊗\mathrm{id}_{X^*}⊗σ^{-1})∘(\mathrm{coev}_X⊗\mathrm{id}_Y) \\ &= (σ⊗\mathrm{id}_1)∘(\mathrm{id}_X⊗\mathrm{ev}_X)∘(\mathrm{coev}_X⊗\mathrm{id}_X)∘(\mathrm{id}_1⊗σ^{-1}) \\ &= σ∘σ^{-1} \end{align*}$$
The point is that the unitors are usually suppressed in the triangle identities.