1. Context
This question came up while reading Passegger's Notes on Turaev-Viro-Barrett-Westbury invariants and TQFT.
Let $\mathbf C$ be a spherical fusion category.
Passegger defines a functor that maps objects $V_1, ..., V_n \in Ob(\mathbf C)$ to $<V_1, ..., V_n> := Hom(I,V_1 \otimes ... \otimes V_n)$. (What it does on morphisms doesn't seem to relevant for the question.)
On page 4 he writes:
2. Question
- Why does above natural isomorphism (compatible with cyclic permutations) exist? What does it look like?
Consider the non-degenerate pairing $\mathrm{Hom} (V,W) \otimes \mathrm{Hom} (W,V) \to \mathbb{k}$ given by $\varphi\otimes\psi \mapsto \mathrm{tr}(\psi\circ\varphi)$. This pairing implies the duality $\mathrm{Hom}(V,W)^\ast \cong \mathrm{Hom}(W,V)$, where the isomorphism is precisely given by the trace. I suggest you draw $\mathrm{tr}(\psi\circ\varphi)$ (recall that this is in fact a map $1 \to 1$) as a string diagram to see what's going on.
Applied to the bracket spaces one therefore gets $\langle V_1 , \ldots , V_n\rangle^\ast \cong \mathrm{Hom} (V_1 \otimes \ldots \otimes V_n, 1)$. Now you appropriately pre-compose a morphism in $\mathrm{Hom} (V_1 \otimes \ldots \otimes V_n, 1)$ with coevaluation maps to arrive at a morphism in $\langle V_n^\ast , \ldots , V_1^\ast \rangle$. Taking evaluation maps at the same places is the inverse (use the snake/zig-zag identity, and the identification of left and right duals by pivotality). Again, string diagrams are your friend. To sum up, we have a functorial isomorphism $\langle V_1 , \ldots , V_n\rangle^\ast \cong \langle V_n^\ast , \ldots , V_1^\ast \rangle$, as desired.
Similar manipulations show that this isomorphism is compatible with cyclic permutations $\langle V_1 , \ldots , V_n\rangle \cong \langle V_n, V_1 , \ldots , V_{n-1}\rangle$. (In my answer https://math.stackexchange.com/a/4260715/389613 I briefly explain how these isomorphisms are obtained from the pivotal structure.)