The objects of the functor category G$-representations and equivariant linear maps}$\textup{\textsf{cat}}\mathbbe{k}^{\mathsf{B} G}G\mathbbe{k}GHG\textup{fun}_H^G \colon \textup{\textsf{cat}}\mathbbe{k}^{\mathsf{B} G} \to \textup{\textsf{cat}}_\mathbbe{k}^{\mathsf{B} H}GH\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 , 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 -sets, -spaces, based -spaces, or indeed -objects in any category, although if the ambient category has few limits and colimits, these adjoints need not exist.