Every small colimit can be constructed as a coequalizer between coproducts over the arrows and objects of the index category

Apply [L1] to $D^{\mathrm{o}\mathrm{p}}:{\mathcal{J}}^{\mathrm{o}\mathrm{p}}\to {\mathcal{C}}^{\mathrm{o}\mathrm{p}}$. By [L2], its object-indexed product becomes $S$, its arrow-indexed product becomes $R$, and the two coordinate maps become $d$ and $c$ with exactly the displayed summand equations.

The equalizer universal property in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ becomes the coequalizer property [F2] in $\mathcal{C}$. The limiting cone equations become $q{\iota }_j=q{\iota }_{k}D(u)$, and existence and uniqueness of mediating arrows both reverse to the colimit clauses.

Thus the coequalizer is a colimit. When $\mathcal{J}$ is empty, both coproducts are initial and the construction returns the initial-object colimit, exactly dual to the boundary case in [L1].