The title pretty much sums it up.

More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ in $D$ satisfies $$ D(\mathrm{colim} \,F, d) \cong [C,D](F, d) $$ for each object $d$ of $D$, where in the right-hand side $d$ denotes the constant functor, and $[C,D]$ the functor category.

This can be seen as a special case of a Kan extension, which satisfies $$ [E,D](\mathrm{Lan}_G F, K) \cong [C,D](F-,K\circ G-) $$ for each functor $K:E\to D$. Namely, by setting $E$ the terminal category we get exactly a colimit.

Just as well, a colimit is a special case of a weighted colimit, which satisfies $$ D(\mathrm{colim}_W \,C, d) \cong [C^{op}, \mathrm{Set}](W-, D(F-, d)) $$ for each object $d$ of $D$. We get an ordinary colimit by setting $W$ to be the constant presheaf at the singleton.

Now, is there a common generalization?

Note that

- In the Kan extension, the "dependent variable" of $F$ is paired to $K\circ G$, while in the weighted colimit, it is paired to $W$. So it's unclear how to fit both dependencies together.
- One can express Kan extensions as particular weighted colimits -
*this is not what I'm asking*.

(I could ask the same question for the enriched case.)

Any reference would also be welcome.