Why are invertible objects reflexive in a tensor category?

Note, that $X^{-1} = X^\vee$ and $ev_X$ corresponds to $\delta$. To see this, notice that by the Yoneda Lemma we have a bijection $$ \operatorname{Nat}(h_{X^{-1}},\operatorname{Hom}(-\otimes X,1)) \longrightarrow \operatorname{Hom}(X^{-1}\otimes X,1), $$ where $h_{X^{-1}}(Y):= \operatorname{Hom}(Y,X^{-1})$. So $\delta\in \operatorname{Hom}(X^{-1}\otimes X,1)$ gives a natural transformation $\tau\colon h_{X^{-1}}\rightarrow \operatorname{Hom}(-\otimes X,1)$. More explicitly, for $Y\in \mathsf C$ we have $$ \tau_Y\colon \operatorname{Hom}(Y,X^{-1}) \xrightarrow{f\mapsto \operatorname{Hom}(f\otimes X,1)(\delta)} \operatorname{Hom}(Y\otimes X,1), $$ i. e. given $f\colon Y\rightarrow X^{-1}$, then $$ \tau_Y(f)\colon Y\otimes X\xrightarrow{f\otimes1_X} X^{-1}\otimes X \xrightarrow\delta 1. $$ It still needs to be checked that $\tau$ is an isomorphism, which is clear since $$ \operatorname{Hom}(Y,X^{-1}) \xrightarrow{f\mapsto f\otimes1_X} \operatorname{Hom}(Y\otimes X, X^{-1}\otimes X) $$ is bijective as $-\otimes X$ is an equivalence of categories (and in particular fully faithful) and $\delta$ is an isomorphism.
With this, $ev_X = \tau_{X^{-1}}(1_{X^{-1}}) = \delta$ is immediate.

From this, it follows that $X = (X^{-1})^{-1} = X^{\vee\vee}$. Now, $i_X\colon X\rightarrow (X^{-1})^{-1}$ corresponds to $$ X\otimes X^{-1} \xrightarrow{\psi_{X,X^{-1}}} X^{-1}\otimes X \xrightarrow\delta 1, $$ i. e. it is the composition of isomorphisms $$ X\xrightarrow{1_X\otimes\delta^{-1}} X\otimes(X^{-1})^{-1}\otimes X^{-1}\xrightarrow{1_X\otimes\psi} X\otimes X^{-1}\otimes (X^{-1})^{-1}\\ \xrightarrow{\psi\otimes1_{(X^{-1})^{-1}}} X^{-1}\otimes X\otimes (X^{-1})^{-1} \xrightarrow{\delta\otimes1_{(X^{-1})^{-1}}} 1\otimes (X^{-1})^{-1} = (X^{-1})^{-1} $$ and thus itself an isomorphism.

A more conceptual reason for this takes the view that a tensor category is simply a 2-category which happens to have only one object. Then duals in the tensor category become adjoints in the 2-category, and inverses of objects in the tensor category are equivalences in the 2-category. Since every equivalence can be refined to an adjoint equivalence, it follows that an inverse is a dual.