Q:

Contruction of Weighted Colimit in a 2-category

On page 306 of Kelly's Elementary Observations on 2-categorical Limits, it is explained that a weighted limit $\{F, G\}$ in a 2-category can be constructed as the equalizer of $v$ and $w$ in
$$
(3.2) \qquad\qquad
\{F, G\} \underset{u}{\longrightarrow}
\Pi_{P \in \mathcal{P}}
\{FP,GP\}
\stackrel{\overset{v}{\longrightarrow}}{\underset{w}{\longrightarrow}}
\Pi_{P,Q \in \mathcal{P}}
\{\mathcal{P}(P,Q) \times FP,GQ\}
$$
What would be the similar formula for a weighted colimit? 

A:

A weighted (co)limit in an arbitrary (co)complete enriched category can be expressed as a (co)end.   The formulas are:$$
F\star G \cong \int^C FC\otimes GC \qquad \{F,G\} \cong \int_C [FC,GC]
$$
where I'm using $F\star G$ as the weighted colimit, $X\otimes C$ as the tensor, and $[X,C]$ as the cotensor. 
If you look in Basic Concepts of Enriched Category Theory, also by Kelly, figure 2.2 (see also figure 3.68), you have the definition of an end as an equalizer:
$$\int_C T(C,C) \longrightarrow \prod_C T(C,C) \stackrel{\longrightarrow}{\longrightarrow} \prod_{C,D}[\text{Hom}(C,D),T(C,D)]$$
If you set $T(P,Q) = [FP,GQ]$, you get the example you started with modulo a little uncurrying on the right. 
The coends are (unsurprisingly) the dual of ends leading to the coequalizer:$$
\coprod_{C,D}\text{Hom}(D,C)\otimes T(C,D) \stackrel{\longrightarrow}{\longrightarrow}\coprod_C T(C,C) \longrightarrow \int^C T(C,C)
$$
where the parallel arrows are $(c,d,f,t) \mapsto (d, T(f,id_d)(t))$ and $(c,d,f,t) \mapsto (c,T(id_c,f)(t))$.  To make sense of these maps in general requires the notion of extraordinary naturality, but for $\mathbf{Cat}$ they can be taken more or less literally.   If you look up the definition of (co)ends as limiting (co)wedges, you may recognize the parallel arrows as being the arms of the (co)wedges.   If we hit the above diagram with $[-,X]$ we get the appropriate diagram verifying the fact that $[\int^{C^{op}} T(C,C),X] \cong \int_C[T(C,C),X]$, that is:
$$
\prod_{C,D}[\text{Hom}(C,D),[T(C,D), X]] \stackrel{\longleftarrow}{\longleftarrow}\prod_C [T(C,C),X] \longleftarrow [\int^{C^{op}} T(C,C),X] \cong \int_C[T(C,C),X]
$$

