Pointwise formula for Kan extensions

Theorem 21.4.3. Let \(F\colon I \to C\) and \(\phi \colon I \to J\) be functors.

(1)

Assume that for every object \(j\) of \(J\) the composite functor \[ I_{/j} \xrightarrow {s} I \xrightarrow {F} C \] admits a colimit in \(C\). Then these colimits assemble into a functor \[ \phi _!(F) \colon J \to C. \] Furthermore, the maps \[ F(i) \simeq \colim _{i' \in I_{/i}} F(i') \to \colim _{i' \in I_{/\phi (i)}} F(i') = \phi _!(F)(\phi (i)) = \phi ^*\phi _!(F)(i) \] assemble into a natural transformation \(\eta _F\colon F \to \phi ^*\phi _!(F)\) exhibiting \(\phi _!(F)\) as a left Kan extension of \(F\) along \(\phi \).

(2)

Dually, if for every \(j\) in \(J\) the composite \[ I_{j/} \xrightarrow {t} I \xrightarrow {F} C \] admits a limit in \(C\), then these limits assemble into functor \[ \phi _*(F) \colon J \to C \] which is a right Kan extension of \(F\).