Left Kan extension as a comma-category colimit ($({\mathrm{Lan}}_{F}Q)(d)={\mathrm{colim}}_{(F\downarrow d)}Q(c)$)

For small categories and a set-valued functor $Q$, each displayed colimit exists in the Category of sets. A representative $(c,u:Fc\to d,x\in Qc)$ maps under a natural transformation $\eta :Q\to GF$ to $G(u){\eta }_c(x)$. The colimit identifications make this well-defined, proving the adjunction between left Kan extension and precomposition.