$\mathcal D$-$\Omega$ duality on the contra side

Given a smooth morphism of schemes $X\rightarrow T$, denote by $\mathcal D_{X/T}^{\mathsf{cr}}$ the sheaf of rings of fiberwise crystalline differential operators on $X$ relative to $T$ and by $\Omega^\bullet_{X/T}$ the de Rham sheaf of DG-algebras of relative differential forms on $X$ over $T$. Assume that the scheme $X$ is quasi-compact and semi-separated. We construct a commutative square diagram of triangulated equivalences between four triangulated categories: the derived category of quasi-coherent sheaves of $\mathcal D_{X/T}^{\mathsf{cr}}$-modules, the reduced coderived category of quasi-coherent DG-modules over $\Omega_{X/T}^\bullet$, the derived category of contraherent cosheaves of $\mathcal D_{X/T}^{\mathsf{cr}}$-modules, and the reduced contraderived category of contraherent DG-modules over $\Omega_{X/T}^\bullet$. The equivalence involving the contraderived category was previously known for affine varieties only; we use contraherent cosheaves in order to obtain a nonaffine generalization of the "contra side" of the story. The exposition is written in the generality of finite locally free twisted Lie algebroids $(\mathfrak g,\widetilde{\mathfrak g})$ over quasi-compact semi-separated schemes $X$, the quasi-coherent twisted universal enveloping quasi-algebras of $(\mathfrak g,\widetilde{\mathfrak g})$, and the Chevalley-Eilenberg quasi-coherent CDG-quasi-algebras of $(\mathfrak g,\widetilde{\mathfrak g})$. The equivalence between the derived categories of quasi-coherent and contraherent $\mathcal A$-modules, called the "naive co-contra correspondence", is proved quite generally for any quasi-coherent quasi-algebra $\mathcal A$ over $X$.