In A duality formalism in the spirit of Grothendieck and Verdier, Boyarchenko and Drinfeld consider a monoidal category $(\mathcal{M}, \otimes, \mathbf{1})$ together with an object $K \in \mathcal{M}$ such that there is an equivalence $D \colon \mathcal{M}^\mathrm{op} \to \mathcal{M}$ where for all $Y \in \mathcal{M}$ the functor $\operatorname{Hom}((-) \otimes Y, K)$ is representable by $DY$.

In this setup they define a new product via $X \odot Y := D^{-1}(DY \otimes DX)$ and claim that this defines a monoidal structure on $\mathcal{M}$. I fail to see what the associator morphism $$D^{-1}(DZ \otimes D^{-1}(DY \otimes DX)) \to D^{-1}( D^{-1}(DZ \otimes DY) \otimes DX)$$ should be. How can one construct such a morphism?

In A duality formalism in the spirit of Grothendieck and Verdier, Boyarchenko and Drinfeld consider a monoidal category $(\mathcal{M}, \otimes, \mathbf{1})$ together with an object $K \in \mathcal{M}$ such that there is an equivalence $D \colon \mathcal{M}^\mathrm{op} \to \mathcal{M}$ where for all $Y \in \mathcal{M}$ the functor $\operatorname{Hom}((-) \otimes Y, K)$ is representable by $DY$.

In this setup they define a new product via $X \odot Y := D^{-1}(DY \otimes DX)$ and claim that this defines a monoidal structure on $\mathcal{M}$. I fail to see what the associator morphism $$D^{-1}(DZ \otimes D(D^{-1}(DY \otimes DX))) \to D^{-1}( D(D^{-1}(DZ \otimes DY)) \otimes DX)$$ should be. How can one construct such a morphism?

Edit: I was missing a $D$ in the domain and codomain. With the correct domain and codomain, one can get an evident associator using that $D^{-1}D \cong \mathrm{Id}$ (cf. the answer of HeinrichD).