Possible Duplicate:
Why is $ U \otimes \operatorname{Ind}(W) = \operatorname{Ind}(\operatorname{Res}(U) \otimes W)$?
I am working on a problem out of Fulton and Harris:
Show that $U \otimes \operatorname{Ind} W \cong \operatorname{Ind}(\operatorname{Res}(U)\otimes W)$
Expanding this out from the left I get to a point where $\bigoplus_{\sigma \in G/H} U \otimes \sigma W$ and by expanding from the right I get to a point where $\bigoplus_{\sigma \in G/H} \sigma \operatorname{Res}_H^G U \otimes \sigma W$.
The only remaining step, if I have done everything correctly thus far, is to show that $\sigma \operatorname{Res}_H^G(U) \cong U$ if $\sigma \in G/H$.
From what I understand, as a vector space $\operatorname{Res} U= U$. The only difference is that as a representation they are different. Thus perhaps it is better to specify where my tensors are being taken: $U \otimes_G \operatorname{Ind} W \cong \operatorname{Ind}(\operatorname{Res}(U)\otimes_{H} W)$ and thus the expansions from the left and right are $\bigoplus_{\sigma \in G/H} U \otimes_G \sigma W$ and $\bigoplus_{\sigma \in G/H} \sigma \operatorname{Res}_H^G U \otimes_H \sigma W$. But such makes me believe I made a mistake since $\sigma \operatorname{Res}_H^G U \otimes_H \sigma W$ shouldn't be defined since $\sigma$ may not be in $H$. Thus it should really be $\bigoplus_{\sigma \in G/H} \sigma (\operatorname{Res}_H^G U \otimes_H W)$.
As you can see I am sort of getting stuck in loops here. Please help!