Unclear asymmetry in Lie-algebra module structure on space of linear transformations Hom(V,W)

Question

Let $L$ be a (finite dimensional) Lie-algebra. Let $V, W$ be finite-dimensional vector spaces. If $V,\; W$ are in addition $L$-modules (see, e.g., 6.1 in Humphreys Introduction to Lie Algebras), then using the isomorphism $V^{*}\otimes W\simeq Hom\left(V,W\right)$, we can define an $L$-module structure on $Hom\left(V,W\right)$ by $x.\left(f\otimes w\right)\left(v\right):=\left(f\otimes x.w+x.f\otimes w\right)\left(v\right)=f\left(v\right)\otimes x.w-f\left(x.v\right)\otimes w$ for $x\in L$, $v\in V$, $f\in V^{*}$. This can be written explicitly as

$x.T\left(v\right)=\sum_{i}x.\left(v_{i}^{*}\otimes Tv_{i}\right)\left(v\right)=\sum_{i}\left[v_{i}^{*}\left(v\right)\; x.Tv_{i}-v_{i}^{*}\left(x.v\right)\; Tv_{i}\right]$

for $T\in Hom\left(V,W\right)$, $\left(v_{i}\right)_{i}$ a basis for V.

Question: The above equation has a clear asymmetry (minus sign) between the action of $L$ on the domain (-) and its action on the codomain (+). Having such an asymmetry is surprising. Technically, it comes from the definition of the $L$-modules structure $\left(x.f\right)\left(v\right):=-f\left(x.v\right)$ for the dual of the $L$-module $V$. Yet, this is merely a technical explanation for the above asymmetry. Why do we have this asymmetry?

Answer 1

If $V$ is a representation of a group $G$ then the dual space $V^*$ is naturally a $G^{\text{op}}$-representation. To get a $G$-action use inversion. For a Lie algebra, the analogue of inversion is $x \mapsto -x$.

To see this fact about groups from a more general perspective: let $BG$ be the groupoid whose one object has automorphism group $G$. Then an action of $G$ on an object $X$ of a category $\mathscr{C}$ is a functor $BG \to \mathscr{C}$ which sends the unique object of $BG$ to $X$. If $F : \mathscr{C}^{\text{op}} \to \mathscr{D}$ is a contravariant functor then we obtain an action $$B(G^{\text{op}}) \to (BG)^{\text{op}} \to \mathscr{C}^{\text{op}} \to \mathscr{D}$$ of $G^{\text{op}}$ on $F(X)$. Now apply this with $\mathscr{C} = \mathscr{D}$ the category of vector spaces and $F$ the duality functor.