The Orbit Functor and Its Right Adjoint

Let us consider a functor $\mathcal{O}:G$$\mathcal O:G$-$\mathbf{Set}\to \mathbf{Set}$$\mathbf{Set}\to\mathbf{Set}$.

For a $G$$G$-Set $X$$X$, $\mathcal{O}(X):=X/G$$\mathcal O(X):=X/G$ is the set of orbits. For an $G$$G$-map $f$$f$, we have $f(gx)=gf(x)$$f(gx)=gf(x)$ hence it maps orbit of $x$$x$ to orbit of $f(x)$$f(x)$.

This functor admits a right adjoint, $\mathrm{\Delta }:\mathbf{Set}\to$$\Delta:\mathbf{Set}\to$$G$$G$-$\mathbf{Set}$$\mathbf{Set}$. For each set $X$$X$, $\mathrm{\Delta }(X)$$\Delta(X)$ is the $G$$G$-Set with trivial action.

$${\mathrm{Hom}}_{\mathbf{Set}}(\mathcal{O}(X),Y)\cong {\mathrm{Hom}}_{G-\mathbf{Set}}(X,\mathrm{\Delta }(Y))$$

You could replace $\mathbf{Set}$$\mathbf{Set}$ by $\mathbf{Top}$$\mathbf{Top}$ to get similar results.