The classical Frobenius reciprocity theorem asserts the following:
If $W$ is a representation of $H$, and $U$ a representation of $G$, then $(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res U})_{H}.$
The proof in the standard textbook (Fulton&Harris, Dummit&Foote,etc) is easy to understand. What puzzled me is this Frobenius theorem that appears in Raoul Bott's paper:
"Proposition 2.1. Let $W$ be a $G$-module, let $M$ be an $H$-module and denote by $i^{*}W$, the restriction of $W$ to $H$. Then, $Hom_{G}(W,\Gamma MG)\cong Hom_{H}(i^{*}W, M).$
In here the $\Gamma MG$ is defined to be the section of the bundle $G\times_{H} M\rightarrow G/H$, with $G\times_{H}M$ defined to be $G\times M/(g,m)\approx (gh,h^{-1}m)$.
Bott claimed in his paper (Homogeneous differential operators) that this isomorphism is quite canoical, yet not only I could not understand his proof, but also I could not see how the isomorphism is anything but canonical. There should be some kind of relationship between this and the classical theorem, but I could not get it as well. After several hours pondering I decided to ask in here as the matter is purely technical.