  We have two parallel goals of 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, we introduce a
generalization for quivers and show that this generalization can be seen as
essentially the same as $n$-representations of quivers. This is significantly
important because it allows us to build a new quiver $\mathscr{Q}_{\!_{(\Qq_1,
\Qq_2,..., \Qq_n)}}$, called $n$-quiver, from any given quivers
$\Qq_1,\Qq_2,...,\Qq_n$, and enable us to identify each category $Rep_k(\Qq_j)$
of representations of a quiver $\Qq_j$ as a full subcategory of the category
$Rep_k(\mathscr{Q}_{\!_{(\Qq_1, \Qq_2,..., \Qq_n)}})$ of representations of
$\mathscr{Q}_{\!_{(\Qq_1, \Qq_2,..., \Qq_n)}}$ for every $j \in
\{1,\,\,2,\,\,..., \,\,n\}$.
