Universal property

Remarks

Equivalently, $(A, u)$ is an initial object of the comma category $(X \downarrow F)$, whose objects are pairs $(A', f \colon X \to F(A'))$; in the dual case it is a terminal object of $(F \downarrow X)$. Since initial objects are unique up to unique isomorphism, so are universal morphisms.

Example: let $U \colon \mathbf{Grp} \to \mathbf{Set}$ send a group to its underlying set, and let $S$ be a set. The free group $F(S)$ with the inclusion $u \colon S \to U(F(S))$ of generators is a universal morphism from $S$ to $U$. Indeed, every function $f \colon S \to U(H)$ to a group $H$ extends to exactly one group homomorphism $g \colon F(S) \to H$ with $g \circ u = f$.

Non-example: take $S = \{s\}$ and the pair $(\Z \times \Z, \ s \mapsto (1, 0))$. The function $s \mapsto 1 \in \Z$ extends to the homomorphisms $(m, n) \mapsto m$ and $(m, n) \mapsto m + n$, so uniqueness fails and this pair is not universal.