  Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras with $\mathfrak{h}$
finite dimensional and consider ${\mathcal A} = {\mathcal A} (\mathfrak{h}, \,
\mathfrak{g})$ to be the corresponding universal algebra as introduced in
\cite{am20}. Given an ${\mathcal A}$-module $U$ and a Lie $\mathfrak{h}$-module
$V$ we show that $U \otimes V$ can be naturally endowed with a Lie
$\mathfrak{g}$-module structure. This gives rise to a functor between the
category of Lie $\mathfrak{h}$-modules and the category of Lie
$\mathfrak{g}$-modules and, respectively, to a functor between the category of
${\mathcal A}$-modules and the category of Lie $\mathfrak{g}$-modules. Under
some finite dimensionality assumptions, we prove that the two functors admit
left adjoints which leads to the construction of universal ${\mathcal
A}$-modules and universal Lie $\mathfrak{h}$-modules as the representation
theoretic counterparts of Manin-Tambara's universal coacting objects
\cite{Manin, Tambara}.
