I am attempting to do a calculation of a colimit in $Cat$, the category of small categories. To this end, people have suggested that I do this by calculating coproducts and using coequalizers. I have no idea how to do this. I have seen the definition for coproducts in Cat but how do we then use coequalizers to find colimits? I am going to have an infinite diagram over which I want to take a colimit to find a particular category. I want to build up my colimit diagram by having a small number of objects in $Cat$, then adjoining one new object to this and finding the new colimit. I can imagine taking successive coproducts, but I am really shakey as to how to do all this. Can anyone give me some direction?

1$\begingroup$ Do you know how to construct a colimit of any diagram via coproducts and coequalizers in arbitrary category? $\endgroup$ – Oskar Jan 23 '16 at 16:07

$\begingroup$ No, I don't. I can read about that, but please give some explanation. $\endgroup$ – Ben Sprott Jan 23 '16 at 16:08

3$\begingroup$ Taking intro account how complicated coequalizers in Cat are, I'd say the suggestion is not very useful. $\endgroup$ – Fernando Muro Jan 23 '16 at 22:07

$\begingroup$ @FernandoMuro, how would you suggest I proceed, then? $\endgroup$ – Ben Sprott Jan 25 '16 at 14:47

$\begingroup$ @BenSprott It depends very much on the shape of your diagram. If it is something as complicated as a double arrow, then the direct approach and the coequalizer approach would be equivalent. But sometimes things are easier, e.g. a pushout diagram where one arrow is free. $\endgroup$ – Fernando Muro Jan 25 '16 at 14:49
Firstly, there exists a general method to construct colimits in arbitrary category via coproducts and coequalizers. I will point it briefly. Let $A$ and $B$ be categories, $T\colon A\to B$ be a functor. If $B$ has coproducts of all families indexed by objects and morphisms of $A$ and all binary coequalizers, then such colimit exists. It is the coequalizer of the pair $(F,G)$, where $F$ and $G$ are morphisms from $\coprod_{f\in Arr(A)}dom(T(f))$ to $\coprod_{a\in Ob(A)}T(a)$, such that $F\circ i_f=i_{dom(f)}$ and $G\circ i_f=i_{cod(f)}\circ T(f)$. This fact is dual to the analogous one about constructing limits via products and equalizers, which you can find in CFWM. It is a general picture.
The category $\mathbf{Cat}$ is cocomplete, i.e for every small category (graph) $A$ and every functor (diagram) $T\colon A\to\mathbf{Cat}$ there exists a colimit of $T$, which one can construct via corresponding coproducts and coequalizers in $\mathbf{Cat}$.
It may be difficult to calculate coequalizers in $\mathbf{Cat}$. Let $A$ and $B$ be categories, $T,S\colon A\to B$ be functors between them. Then the coequalizer of the pair $(T,S)$ is $B/C$, where $C$ is the free congruence generated by $T(a)=S(a)$ and $T(f)=S(f)$ for any object $a\in A$ and any morphism $f\in A$. See also: Generalized congruences  Epimorphisms in Cat.

$\begingroup$ The "congruence" in your last paragraph is more general than the usage that I for one am familiar with  usually we don't allow identification of objects in a congruence, but only of morphisms with the same domain / codomain. When you allow yourself to identify objects, things get more complicated because new pairs of morphisms become composable and you have to freely adjoin new morphisms to represent their composite. For example, the coequalizer of the two inclusions of the terminal category into the walking arrow is the monoid $\mathbb{N}$ considered as a 1object category. $\endgroup$ – Tim Campion♦ Jan 23 '16 at 19:37

$\begingroup$ @TimCampion Yes, it's worth emphasizing that $C$ is actually a generalized congruence (by the terminology from the link): an equivalence relation on objects together with a partial equivalence relation on sequences of morphisms (which are in some sense compatible). $\endgroup$ – Oskar Jan 23 '16 at 20:28