Moyal-Weyl deformations of $\mathrm{DGA}$ and $\mathrm{DGCA}$

We consider a natural variant of the Moyal-Weyl product and show that it yields a functorial deformation of differential graded algebras and that we can deform coalgebras in a similar way. The Moyal-Weyl deformation of graded algebras has already been introduced by Fedosov \cite{Fe} and also appears in \cite{Ge} as part of an $A_\infty$ structure, but we are not aware that the analogous result for differential graded coalgebras already appeared in the literature and discuss the Moyal-Weyl of the category of complexes as an example.