Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$. A $\textbf{universal arrow}$ over $x \in \mathcal B$ is a morphism $u:x\to U(c)$ that has the following universal property: for each morphism $f: x\to U(b)$ there exists a unique morphism $f' : c\to b$ such that $U(f') \circ u = f$.
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 $\mathcal{C}$ be the non-full subcategory of $\mathcal{L}\mathrm{at}_{\lor}^{\infty}$ whose objects are suplattice and morphisms are meet- and join-preserving maps. Then in the construct $\mathcal{C}$, there exists a free object over a set $X$ if and only if the cardinality of $X$ is not greater than 2, i.e., $\lvert X \rvert \leq 2$.