If $F:C \to D$ is functor between categories $C$ and $D$ and $X$ is an object of $D$, then a universal map from $X$ to $F$ is a pair $(A_X,u)$ where $A_X \in \mathrm{ob}(C)$ and $u \in D(X,F(A_X))$ such that if $A$ is any object of $C$ and morphism $f\in D(X,F(A))$ there is a unique morphism $\tilde{f}\in C(A_X,A)$ such that $f = F(\tilde{f})\circ u$.
If $(A, u: X \to F(A))$, $(A', u': X \to F(A'))$ - two universal maps from $X$ to $F$, then $A$ and $A'$ are isomorphic in category C. Are they isomorphic up to unique isomorphism? I mean, may so happen that we can find more then one isomorphism between $A$ and $A'$ in category $C$?