Categorification of commutative sum In the introduction to locally presentable categories at nLab, it is stated:

Now, the categorification of “commutative sum” is colimit.

My intuition in that context would be that the categorification of "commutative sum" is coproduct. It was clear for me that "adding elements" might be put in parallel with "taking a coproduct of objects". However, how should I interpret coequalisers or other kind of non-coproduct colimits?
 A: The analogy just doesn't make that much sense yet, at this point. You need to see it taken farther first: 
Taking the free abelian group on a set categorifies to freely completing a category under colimits; this is the Yoneda embedding $C \to [C^{op}, \text{Set}]$. Note that you can think of a presheaf $F : C^{op} \to \text{Set}$ as a "vector" whose "components" are the values of the presheaf, and it is possible to make sense of the statement that a presheaf is the "sum" of its components in the colimit sense, by writing a presheaf as a colimit of representable presheaves. 
Next, given two finite sets $X, Y$ we can consider linear maps $\mathbb{Z}[X] \to \mathbb{Z}[Y]$ from the free abelian group on $X$ to the free abelian group on $Y$. It's not hard to see that these correspond to $|X| \times |Y|$ (or is it the other way around?) matrices with entries in $\mathbb{Z}$.
This categorifies as follows. The category of cocontinuous functors $[C^{op}, \text{Set}] \to [D^{op}, \text{Set}]$ is, by the universal property of the Yoneda embedding, equivalent to the category of functors $C \times D^{op} \to \text{Set}$. These are called profunctors or bimodules, and they are intuitively "matrices of sets." Composition is given by an operation which generalizes tensor product of bimodules and which you can think of as categorifying matrix multiplication, but where the sum is a colimit, or more precisely a coend. 
For more along these lines see this blog post, which replaces $\text{Set}$ with $\text{Mod}(k)$ for $k$ a commutative ring but is otherwise about the same thing. 
Also keep in mind that categorifications are not unique. You could choose to use only coproducts as your categorification but you can do a lot more if you use colimits instead. 
