My question arose while studying something about Kan Extensions.
We know that we have the following diagram
$$ \begin{array}{ccc} &&\mathsf{\Delta} & \xrightarrow{\mathcal{F}} & \mathsf{Top}\\ &&\mathcal{y} \searrow& & \nearrow Lan_{\mathcal{y}\mathcal{F}}\\ &&& \mathsf{sSet}, &&&& \end{array} $$ where $Lan_{\mathcal{y}\mathcal{F}}$, stands for the left Kan Extension of $\mathcal{F}$ along the Yoneda Embedding $\mathcal{y}$, of the simplex category $\mathsf{\Delta}$ into the category of its presheaves. Now, because the category of topological spaces is co-complete and the simplex category is small we know the left Kan Extesnsion exists. However for $\mathcal{F}$ being the functor $$[n] \mapsto |\Delta^n|,$$ we get a rather well-known functor denoted by $|-|$, called geometric realization, which is left adjoint of the singularization functor, $$ S : \mathsf{Top} \rightarrow \mathsf{sSet} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, Y \mapsto S_Y = \mathsf{hom}_\mathsf{Top}( \mathcal{F}[-],Y).$$ Now, because of the particular structure these categories possess, means that the right Kan Extension of $\mathcal{F}$ along the Yoneda Embedding $\mathcal{y}$ does exist as well, and moreover we have a certain adjunction of the form $Lan_{\mathcal{y}} \dashv - \circ \mathcal{y} \dashv Ran_{\mathcal{y}}.$ With all the above it is clear what the left Kan extension is, however I'm not sure about the right one. Can you explain what is this functor? (If I haven't done something wrong so far :))
Also for the first adjunction I mentioned, $|-| \dashv S$, is there any categorical theoretic insight (an argument arising from Kan Extension theory for instance) that imposes these functors being adjoints?