Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$. A $\textbf{universal arrow}$ over $x\in \mathcal B$ is a structured arrow $u:x\to U(c)$ with domain $x$ that has the following universal property: for each structured arrow $f: x\to U(b)$ with domain $x$ there exists a unique morphism $\underline{f} : c\to b$ such that $\underline f\circ u=f$.
Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$. A $\textbf{free object}$ over $x\in \mathcal B$ is an object $c\in \mathcal C$ such that there exists a universal arrow $(u,c)$ over $x$.
Theorem: Let $(\mathcal{C},U)$ be a construct such that $U$ is representable by an object $x$. Then for any set $I$ and any object $d\in \mathcal{C}$ the following conditions are equivalent: \begin{enumerate} \item $d$ is a free object over $I$. \item $d$ is an $I$-th copower of $x$. \end{enumerate}