The objects of the functor category $, of $G$-representations and equivariant linear maps}$\textup{\textsf{cat}}\mathbbe{k}^{\mathsf{B} G}$ are $G$-representations over the field $\mathbbe{k}$, and arrows are $G$-equivariant linear maps. If $H$ is a subgroup of $G$, restriction $\textup{fun}_H^G \colon \textup{\textsf{cat}}\mathbbe{k}^{\mathsf{B} G} \to \textup{\textsf{cat}}_\mathbbe{k}^{\mathsf{B} H}$ of a $G$-representation to an $H$-representation is simply pre-composition by the inclusion functor $\mathsf{B} H \hookrightarrow \mathsf{B} G$. This functor has a left adjoint, called induction, and a right adjoint, called coinduction. \(\vcenter{\xymatrix{\cat{Vect}_\mathbbe{k}^{\mathsf{B} G} \ar[r]\mid{\mathrm{res}_H^G} & \textup{\textsf{cat}}_\mathbbe{k}^{\mathsf{B} H} \ar@/^1.5pc/[l]^{\mathrm{coind}_H^G} \ar@/_1.5pc/[l]_{\mathrm{ind}_H^G} \ar@{}[l]^*+{\labelstyle{\perp}}_*+{\labelstyle\perp} }}\) By Proposition \ref{prop:kan-adjoint}, the induction functor is given by left Kan extension along the inclusion $\mathsf{B} H \hookrightarrow \mathsf{B} G$, while coinduction is given by right Kan extension. The reader unfamiliar with the construction of induced representations need not remain in suspense for very long; see Theorem \ref{thm:Kan-formula} and Example \ref{ex:repKan2}. Similar remarks apply for $G$-sets, $G$-spaces, based $G$-spaces, or indeed $G$-objects in any category, although if the ambient category has few limits and colimits, these adjoints need not exist.