= Left Kan extension as a comma-category colimit
{title2=$(\operatorname{Lan}_FQ)(d)=\operatorname{colim}_{(F\downarrow d)}Q(c)$}
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.
Back to article page
