If $G$ is a Lie group with product $\circ: G \times G \to G$, an "obvious" Lie group structure present on the tangent bundle $T G$ is given by taking the differential of the Lie group product $d\circ: TG \times TG \to TG$ (where we have identified $TG \times TG$ and $T(G\times G)$. If $(X, x),(Y,y) \in TG$, then the product on the tangent bundle is given by:
$$(X, x)d\circ (Y, y) = (XY, d(L_X)_Yy + d(R_Y)_X x.$$
This is not the only group structure you can place on the tangent bundle (there's also the product that corresponds to a translation and addition of the Lie algebra elements, for example), but the relation to the group product makes it arguably the most natural.
Now consider the cotangent bundle $T^*G$. If $(X, \lambda),(Y,\nu) \in T^*G$, then the corresponding "obvious" Lie group product here is:
$$(X, \lambda)(Y, \nu) = (XY, d(L_{X^{-1}})_Y^* \nu + d(R_{Y^{-1}})_X^* \lambda.$$
It is not clear to me to motivate that this is the "correct" structure though, as it's not obvious how we can can relate the group product to a product map on the cotangent bundle, as the dual map has the form $d\circ^* : T^*G \to T^*G \times T^*G$.
Can anyone motivate this as the "correct" group structure for me please?
