Example of categorification of a cluster algebra

We present here two detailed examples of additive categorifications of the cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a simple Lie group. The first one is of simply-laced type ($A_3$) and relies on an article by Gei\ss, Leclerc and Schr\"oer. The second is of non simply-laced type ($C_2$) and relies on an article by the author of this note. This is aimed to be accessible, specially for people who are not familiar with this subject.