Isomorphism between Weyl and Cartan models as Hom-Tensor Adjunction

Question

Let $M$ be a manifold, $\Omega$ be the de Rham complex of $M$. Let $G$ is a compact Lie group acting on $M$, $\mathfrak g$ its Lie algebra and $W(\mathfrak g) = \Lambda(\mathfrak g^*) \otimes S(\mathfrak g^*)$ is the Weyl algebra. One of the steps in constructing Cartan's model of equivariant cohomology is the isomorphism $$ (W(\mathfrak g) \otimes \Omega)_{\text{hor}}\cong S(\mathfrak g^*) \otimes \Omega $$ of graded algebras. Can we look at this isomorphism as an example of a hom-tensor adjunction as follows? First, we interpret the horizontal complex as $$ (W(\mathfrak g) \otimes \Omega)_{\text{hor}} \cong \mathrm{Hom}_{\Lambda(\mathfrak g)}(\mathbb R, W(\mathfrak g) \otimes \Omega), $$ where $X \in \mathfrak g$ acts by substitution $\iota_X$ on $W(\mathfrak g) \otimes \Omega$. In other words, we consider abelian graded Lie algebra as vector space isomorphic to $\mathfrak g$ and sitting in degree $-1$, the horizontal complex is the invariants of the algebra and $\Lambda(\mathfrak g)$ is the universal enveloping algebra. Then $$ \mathrm{Hom}_{\Lambda(\mathfrak g)}(\mathbb R, W(\mathfrak g) \otimes \Omega) \cong \mathrm{Hom}_{\Lambda(\mathfrak g)}(\mathbb R, \mathrm{Hom}_{\mathbb R}(\Lambda (\mathfrak g), S(\mathfrak g^*) \otimes \Omega)). $$ The adjunction gives $$ \mathrm{Hom}_{\Lambda(\mathfrak g)}(\mathbb R, \mathrm{Hom}_{\mathbb R}(\Lambda (\mathfrak g), S(\mathfrak g^*) \otimes \Omega)) \cong \mathrm{Hom}_{\mathbb R}(\mathbb R \otimes_{\Lambda (\mathfrak g)} \Lambda (\mathfrak g), S(\mathfrak g^*) \otimes \Omega) \cong S(\mathfrak g^*) \otimes \Omega. $$ Is the above reasoning correct? Is it written somewhere?