Induction and coinduction of permutation representations as Kan extensions

In the one-object setting, an object of $(i\downarrow \ast )$ is just an element $g\in G$. A morphism from $g$ to $g^{\mathrm{\prime }}$ is an element $h\in H$ with $g=g^{\mathrm{\prime }}i(h)$ by the comma-category equation, equivalently $g^{\mathrm{\prime }}=gi(h^{-1})$. So the indexing category is the action groupoid for the right $H$-action on $G$, and the induced diagram on $\mathbf{S}\mathbf{e}\mathbf{t}$ sends the arrow $g\to gi(k)$ to the map $x\mapsto k^{-1}x$ on $X$. By [F2], its colimit is the quotient of $G\times X$ by $(g,x)\sim (gi(k),k^{-1}x)$, equivalently by $(gi(h),x)\sim (g,hx)$, namely $G{\times }_{H}X$. Therefore [L1] identifies the induced representation with the left Kan extension.

Dually, an object of $(\ast \downarrow i)$ is again an element of $G$, and a cone to a set $Y$ is exactly a family of maps indexed by $G$ that is equivariant for the $H$-action. By [F2], the limit is therefore the set of $H$-equivariant maps $G\to X$, written ${\mathrm{Map}}_H(G,X)$. Hence [L1] identifies the coinduced representation with the right Kan extension.