A Universal Investigation of $n$-representations of $n$-quivers

noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$, called an $n$-quiver, and identify each category $Rep_k(mathit{Q}_j)$ of representations of a quiver $mathit{Q}_j$ as a full subcategory of the category $Rep_k(mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}})$ of representations of $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$ for every $j in {1,2,ldots , n}$.