Probably this is a silly question, but I'm going a bit confusing reading some definitions in Adamek's The Joy of Cats.
In essence the problem is given by the uniquess assumption in the definition of functor which lifts limits uniquely. In the whole book, Adamek makes a clear distinction when talking of uniqueness: in fact, uniqueness up to isomorphisms is called essential uniqueness (as in the case of limits, which are "essentially unique"), and leaves the term "unique" to denote the existence of exactly one object/morphism and so on.
Now, suppose I have a functor $F: \mathfrak{A} \to \mathfrak{B}$ which lifts limits as in Definition 13.17. I'd like to prove that $F$ lifts limits uniquely. To this aim, knowing the existence of a limit $\mathcal{L}$ as in Definition 13.17, I should take a second limit $\mathcal{M}$ lifting the starting limit and prove it agrees $\mathcal{L}$. Nevertheless, I'm able to show that $\mathcal{M}$ is isomorphic to $\mathcal{L}$. May I conclude that $F$ lifts limits uniquely by invoking the essential uniqueness of the limit $\mathcal{L}$?
The answer is no.
Proposition 13.21 asserts that $F$ lifts limits, then it does so uniquely if and only if $F$ is amnestic. In fact, the proof of the proposition shows that being amnestic is equivalent to $F$ lifing uniquely limits of singleton diagrams. In particular, essential uniqueness of limits does not imply their unique lifting, as then all functors would be amnestic.