The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra.
Now, we said that due to $Ad_{g_1g_2}(x)=Ad_{g_1}Ad_{g_2}(x)$ we see by conjugating this equation that the above written group action is actually a right-action.
The thing is that our teacher told us now that $Ad^*:G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*,(g,x) \mapsto Ad^*_{g^{-1}}(x)$ is a left-action. Unfortunately, I don't see why this is the case. Could anybody here try to explain why we have now a left-action?
In general, if you have a right action $R : (g,x) \mapsto g.x$, then you can define a left action by $L : (g,x) \mapsto g^{-1}.x$ (and reciprocally).
This comes from the fact that the inverse is an anti-automorphism of G : $(gh)^{-1}=h^{-1}g^{-1}$.