Corestriction after transfer multiplies by the index

Put $u(x)=x$. For each $t\in T$ define the alternating vertex prism $P_{t,n}[(g_0,\dots ,g_n)\otimes m]={\sum }_{i=0}^n(-1{)}^i[(r(t{g}_0),\dots ,r(t{g}_i),t{g}_i,\dots ,t{g}_n)\otimes tm]\mathrm{.}$ Expanding the vertex-deletion differential cancels every face away from the switch in pairs; the two endpoint faces that survive give $d{P}_t+P_{t}d=u_{t,\ast }-r_{t,\ast },$ where $u_{t,\ast }$ uses $(t{g}_0,\dots ,t{g}_n)$ and $r_{t,\ast }$ is the $t$-summand of corestriction after the transfer in [F2]. If $tx=h_t{t}_x$ with $h_t\in H$ and $t_x\in T$, then $t\mapsto t_x$ permutes $T$, $r(txg)=h_{t}r(t_{x}g)$, and both vertex strings and the coefficient $txm=h_t{t}_{x}m$ change by the same diagonal translation. Hence $P={\sum }_{t\in T}{P}_t$ descends to $G$-coinvariants. Equal adjacent input vertices make every prism summand degenerate, so it also descends to normalized chains. Thus $dP+Pd={\sum }_{t\in T}{u}_{t,\ast }-i_{\ast }{\mathrm{Tr}}_T\mathrm{.}$

Each translated summand is $[(g_0,\dots ,g_n)\otimes m]$ in diagonal G-coinvariants. The sum is therefore $[G:H]$ times that chain in every degree, including zero. Homotopic chain maps give the same map on homology because their difference on a cycle is a boundary. Thus $i_{\ast }\mathrm{Tr}=[G:H]\mathrm{id}$.