problems/0034.md

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}