Exercise 1, § 1

Let $(Q_i)_{i\in I}$ be a family of quivers. Denote by S the sum set of the family (Som(Q$_i$)), F the sum set of the family (Fl(Q$_i$))$,o$ and $t$ the mappings from F into S deduced from the origin and term mappings of the quivers $Q_i$. (a) Prove that $Q = (S,F, o, t)$ is a quiver. This is called the sum quiver of the family $(Q_i)$.

(b) For $i\in I$, there exists a unique morphism of quivers $j_i$ from $Q_i$ into Q such that the mapping Som($j_i$) is the canonical mapping of Som(Q$_i$) into S and the mapping Fl($j_i$) is the canonical mapping of Fl(Q$_i$) into S.

(c) Let K be a quiver and, for every $i\in I$, let $\varphi_i$ be a morphism of quivers from $Q_i$ into K. Prove that there exists a unique morphism of quivers $\varphi$ from Q into K such that one has $\varphi \circ j_i=\varphi_i$ for every $i\in I$.