at.algebraic topology - Maurer-Cartan and representable functors on differential graded commutative algebras

Let $mathfrak{g}$ be a differential graded Lie algebra on a charcteristic zero field, whose underlying chain complex is bounded below and degreewise finite dimensional. Then $mathfrak{g}$ defines a Set-valued functor on differential graded commutative algebras (also, with some finiteness and boundedness assumption) mapping each cgda $Omega$ to the set $MC(Omegaotimes mathfrak{g})$ of Maurer-Cartan elements of the dgla $Omegaotimes mathfrak{g}$ (with the natural dgla structure on the tensor product of a dgla with a cdga). It is well known that this functor is representable: $MC(Omegaotimes mathfrak{g})cong Hom_{dgca}(CE(mathfrak{g}),Omega)$, where the Chevalley-Eilenberg dgca $CE(mathfrak{g})$ is the free graded commutative algebra on the shifted linear dual of $mathfrak{g}$ endowed with the differential induced by the dgla structure on $mathfrak{g}$.

If one looks at the category of commutative graded algebras instead (i.e., one forgets the differential), then $CE(mathfrak{g})$ represents the functor $Omegamapsto (Omegaotimesmathfrak{g})^1$.

My question is: is this latter functor representable also in the category of differential graded commutative algebras? if yes, by which dgca?

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