The presentation of associative algebras in terms of generators and relations are very useful as they often give a simple description of a large, potentially infinite dimensional algebra using just a few generators and relations. For example, consider the algebra $\mathcal{A}$ over $\mathbb{C}$ generated by two elements $p,f$ satisfying $$p^n=1, f^n=0, pf=\omega fp,\tag{1}$$ where $\omega$ is a primitive $n$-th root of unity $\omega^n=1$. As we can see, though $\mathcal{A}$ is $n^2$ dimensional, it can be described by just 2 generators and 3 relations.
Do coalgebras also have an analogous description in terms of generators and relations? In particular, the dual $\mathcal{A}^*$ of the above algebra $\mathcal{A}$ is naturally a coalgebra, does $\mathcal{A}^*$ have a simple description in terms of generators and relations?