It is well known that given a pushout and an initial object, coproducts and coequalizers may be constructed. Here, one can find a well given explaination to do so:
Constructing coequalizers via pushouts and initial objects
By the answer given here, it seems that arbitrary coproducts may be constructed via pushouts and initial objects.
Now, Borceux (Handbook of Categorical Algebra, vol. 1, prop. 2.8.2, page 62) shows that a category is finitely cocomplete if and only if it admits an initial object and a pushout (actually, I dualized the corresponding statement). So, I deduce the existence of examples where the above construction fails to be true for arbitrary indexed sets families of objects! Can you provide some of these examples, please? Moreover, does the category have to satisfy other conditions in order to construct arbitrary coproducts via pushouts?
Best Answer
The most basic example is the category of finite sets. It has all finite colimits, but not all colimits (for instance, the diagram of all finite subsets of $\mathbb{N}$ has no colimit).
Another (very similar) example is the category if finitely presentable $R$-modules (where $R$ is any non-trivial ring). It has finite colimits, but not all colimits.
Have a look at the concept of a wide pushout. These are colimits indexed by graphs which are rooted trees of height $1$. The usual case of a pushout is of the tree with two children:
$\begin{array}{ccc} \bullet & \leftarrow & \bullet & \rightarrow & \bullet \end{array}$
But you can also have more edges:
$\begin{array}{ccc} & &\bullet && \\ && \uparrow && \\ \bullet & \leftarrow & \bullet & \rightarrow & \bullet \end{array}$
or
$\begin{array}{ccc} & &\bullet && \\ && \uparrow && \\ \bullet & \leftarrow & \bullet & \rightarrow & \bullet \\ && \downarrow && \\ & &\bullet && \end{array}$
So diagrams of this shape consist of morphisms $(S \to X_i)_{i \in I}$, and the wide pushout $(X_i \to P)$ is the universal family of morphisms making all $S \to X_i \to P$ equal.
A category is cocomplete if it has an initial object and all wide pushouts. In fact, you already know that it has coequalizers, and wide pushouts along the initial object provide arbitrary coproducts, and with coequalizers and coproducts all colimits can be built.