Free object

Remarks

A free object on $X$ is unique up to a unique isomorphism compatible with $i$. If every set has a free object $F(X)$, then $F$ is a functor and is left adjoint to $U$.

Example: in the category of groups, the free group $F(X)$ of reduced words in $X \cup X^{-1}$, with $i$ sending $x$ to the one-letter word, is a free object. Given $f \colon X \to U B$, the morphism $g$ sends a reduced word $x_1^{\epsilon_1} \cdots x_n^{\epsilon_n}$ to $f(x_1)^{\epsilon_1} \cdots f(x_n)^{\epsilon_n}$, and it is the only homomorphism extending $f$.

Non-example: the group $\Z/2\Z$ with $i$ sending the point of $X = \set{x}$ to $1$ is not free on $X$. For $B = \Z$ and $f(x) = 1$, there is no homomorphism $g \colon \Z/2\Z \to \Z$ with $g(1) = 1$, since $2 g(1) = g(0) = 0$.