Can anyone outline the steps for constructing general colimits?

Question

Specifically, in a cocomplete category how can one construct general colimits via other colimits such as initial objects,Coproducts and Coequalizers? (I would prefer not very heavy mathematical notation if possible....)

Answer 1

A good way to approach this question is to look at a generic example, like $\Bbb{Set}$.

Perhaps the dual statement is even easier to visualize: so let us try to express any limit by means of products and an equalizer.

If a diagram (i.e. a functor from a small category $\Delta$), $\ D:\Delta\to \Bbb{Set}\ $ is given, its limit in $\Bbb{Set}$ is the subset of the product $$L:=\left\{(a_x)_{x\in Ob\Delta}\,\mid\, \forall \alpha:x\to y\ \text{ in }\Delta\ \left(a_y=D\alpha(a_x)\right) \right\}\quad \subseteq\ \prod_{x\in Ob\Delta} D(x)$$ Now, for each arrow $\alpha:u\to v\,$ in $\Delta$, we indicated two functions $\prod_{x\in Ob\Delta}D(x)\to D(v)$ in the defining equation: $$ f_\alpha:=(a_x)_x\,\mapsto a_v\ \ \text{ and }\ \ g_\alpha:=(a_x)_x\,\mapsto D\alpha(a_u)\,.$$ These altogether define two maps between products: $$ \prod_{x\in Ob\Delta} D(x) \ \underset{g}{\overset{f}\rightrightarrows}\ \prod_{\matrix{\alpha\in \Delta,\\ \alpha:u\to v}} D(v)\,.$$ And, the limit subset $L$ can be given as their equalizer.

This idea with the final setup can be fully formalized for the general case (using projections and the product property), and you can verify that the resulting equalizer will indeed satisfy the limit property for the diagram $D$.

By dualizing the general proof, we get the dual statement for colimits. I suggest to write it up as well in $\Bbb{Set}$.

Note also that if $\Delta$ is finite, then all products involved are finite.