Let $\mathsf D \overset{K}{\leftarrow} \mathsf C \overset{F}{\to} \mathsf E$ be functors. Suppose we define a pointwise left Kan extension of $F$ along $K$ on objects by the colimit of the composite $K_{/d}\to \mathsf C\to \mathsf E$.
What is a (slick) way to prove the functoriality of this assignment? Particularly, how can we write a pointwise left Kan extension as a composite of functors?
Using the fact that $$ \text{Lan}_KF(d) \cong \int^{c\in\sf C} \hom(Kc,d)\cdot Fc $$ whenever this colimit exists, seems to be the slickest way. More generally, if $F : {\sf C}\to {\sf E}$ is a diagram of $\cal V$-categories and $W : {\sf C}^\text{op}\times{\sf D} \to \cal V$ is a weight, the weighted colimit $d\mapsto [W\otimes F](d)$ is the coend $$ \int^{c\in \sf C} W(c,d)\cdot Fc $$ and in both cases this is functorial in the weight $W$, in the diagram $F$, and in the argument $d$ as a simple consequence of the functoriality of coends.