Uniqueness of Duals in a Monoidal Category

They are always isomorphic, and by a unique isomorphism that preserves the structure of the duality (see this question for the unicity).

Suppose $X^*$ and $\hat{X}^*$ are two different left duals (with corresponding evaluations $\epsilon, \hat\epsilon$ and coevaluations $\eta, \hat\eta$). Define $f : \hat{X}^* \to X^*$ as the composite $$f = (\hat{\epsilon} \otimes 1) \circ (1 \otimes \eta) : \hat{X}^* \xrightarrow{1 \otimes \eta} \hat{X}^* \otimes X \otimes X^* \xrightarrow{\hat\epsilon \otimes 1} X^*$$ Similarly define $\hat{f} = (\epsilon \otimes 1) \circ (1 \otimes \hat{\eta})$. (In both cases I left out the isomorphisms $I \otimes Y \cong Y \cong Y \otimes I$ and the associators, but if you wanted to be 100% precise you should include them). Then it is a tedious but completely mechanical exercise to check that $f$ and $\hat{f}$ are inverse to each other, because of the two axioms (applied respectively to $X^*$ and $\hat{X}^*$) in the Wikipedia article you cited.