I have been wondering for a while about the following question without getting anywhere:
Let $G$ be a group, $N$ a normal subgroup. Can the quotient group $G/N$ be seen as the (category theoretical) colimit of a diagram? If it can, of what diagram?
It's the diagram $N \rightrightarrows G$ where one of the arrows is the inclusion and the other arrow is the zero map. (If $N$ is not necessarily normal you will instead get the quotient by the normal closure of $N$.) More generally, see coequalizer and cokernel.