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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 $\textbf{Set}$?

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    According to [wikipedia](https://en.wikipedia.org/wiki/Limit_(category_theory)#Colimits), a colimit _must_ be universal to be called a colimit. What definition do you have that let you speak of "the" colimit, without implying that it is universal? – 2017-02-14
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    No, I didn't mean universality by that. The universality I meant is the following: Let $F:\mathscr{D}\rightarrow \mathscr{C}$ be a covariant functor with $\mathscr{C}$ have pullbacks. When $(M,\{t_D\})$ is the colimit on $F$ and $f:N\rightarrow M$ is a morphism, let $(GD,s_D,r_D)$ be the pullback of $(f,t_D)$. Then, $G:\mathscr{D}\rightarrow \mathscr{C}$ is a well-defined covariant functor and $(N,s_D)$ is a cocone on $G$. Construct as above, if this cocone is the colimit of $G$ for every $f$, we say the colimit $(M,\{t_D\})$ is universal. – 2017-02-14
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    A less ambiguous term is "stable under pullbacks" as referenced [here](https://ncatlab.org/nlab/show/universal+colimit) and described [here](https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits#ColimitsStableByBaseChange). – 2017-02-14

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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 $G$.