Induction and restriction satisfy the projection formula on character rings

For an honest pair of characters $\chi$ and $\psi$ and any $g\in G$, [F1] gives ${\mathrm{Ind}}_H^G(\chi {\mathrm{Res}}_H^G\psi )(g)=\frac{1}{\mathrm{\mid }H\mathrm{\mid }}{\sum }_{x\in G,x^{-1}gx\in H}\chi (x^{-1}gx)\psi (x^{-1}gx)$.

Since $\psi$ is a class function by [F3], $\psi (x^{-1}gx)=\psi (g)$ for each summand of step 1.1. Factoring that constant out of the finite sum and applying [F1] again yields ${\mathrm{Ind}}_H^G(\chi {\mathrm{Res}}_H^G\psi )(g)=\psi (g){\mathrm{Ind}}_H^G\chi (g)=(({\mathrm{Ind}}_H^G\chi )\psi )(g)$.

So the identity holds for ordinary characters. By [F4], both induction and multiplication extend $\mathbb{Z}$-bilinearly to virtual characters, so the same pointwise identity holds for all $\chi \in R(H)$ and $\psi \in R(G)$.

This pointwise equality is the projection formula in $R(G)$, and [F2] identifies the pointwise product on the right with the character-ring product coming from tensor products. The adjoint viewpoint from [F5] is compatible with it, but step 3.1 already proves the formula.