Let $\mathscr{C}$ be a small category and $F:\mathscr{C}\rightarrow \textbf{Set}$ be a covariant functor. I understood the proof given in my text showing that "the colimit of $F$ is universal when $\mathscr{C}$ is a discrete category or is the category $\{\bullet\rightrightarrows \bullet\}$". Right after this assertion, the author concludes that this implies every small colimit in $\textbf{Set}$ is universal, but I don't get this.
I know that if a category has coequalizers and coproducts, then this category is cocomplete. However, I don't get why it is sufficient prove the universality for the case of coequalizers and coproducts, since the definition of universality is given in terms of pullbacks.
How do I derive the universality of colimits in $\textbf{Set}$?
This is just using the usual construction of the colimit as a coequalizer of coproducts of values. If $$M=\mathrm{coeq}(\coprod_f F(\mathrm{Dom}(f))\stackrel{\to}{\to}\coprod_c F(c))$$ then $N$ is the coequalizer of the pullback along $f$ of the two terms, which are the coproduct of copies of $F(\mathrm{Dom}(f))\times_M N,F(c)\times_M N$. Thus reduced, $N$ becomes the coequalizer of the standard diagram for your pulled back functor $G$.