"unit" morphism in a closed symmetric monoidal category is an isomorphism

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I'm learning about symmetric monoidal closed categories from Borceux's handbook and found myself asking a question similar to this already asked question :

Morphisms in a symmetric monoidal closed category.

However, I'm struggling to understand the accepted answer, nor do I understand why $$\mathrm{ev}_{IC} \circ \rho_{[I, C]}^{-1} \circ i_c =1$$

Is this some basic fact about adjunctions that I'm missing?Any explanation would be helpful.