The problem concerns the Gabriel-Zisman localization of a category $C$ and a set $S$ of morphisms in it. By definition it’s a functor $f$ from $C$ to $D$ which satisfies the following universal property: For any category $A$, the Functor $F$ by composition of $f$ from Fun($D$, $A$) to Fun($C$, $A$) is fully faithful and its essential image is all functors from $C$ to $A$ that send elements of $S$ to isomorphism in $A$.
Here are my problems.
I know if $D$ exists, then it is unique up to a equivalence, and the equivalence is unique up to an isomorphism. But why is this isomorphism unique? Or what does it mean the equivalence is unique up to a unique isomorphism?
Can you please tell me how to solve them? Maybe a hint or a idea.
To answer your question about the meaning of the claim, it means that it $f: C\to D, g: C\to E$ are localizations at $S$, and $h_1,h_2: D\to E$ are equivalences with isomorphisms $\theta_i : h_i\circ f \simeq g$, then there is an isomorphism $\eta : h_1\implies h_2$, and this isomorphism is unique to be "the identity when restricted to $C$".
Here what I mean by "being the identity when restricted to $C$" is as follows : for an object $c$ of $C$, you have $h_1(f(c))$ and $h_2(f(c))$. They may be different, but they are isomorphic, and there is a given isomorphism that "identifies them", and that we may treat as the identity if we think of the two as being equal. This isomorphism is the composite $\theta :h_1\circ f \implies g \implies h_2\circ f$.So what we're saying is that $\eta$ is unique in satisfying $\eta f = \theta$ (where $\eta f (x) = \eta_{f(x)}$).
So it's like for, say, categorical products: we're not saying there is a unique isomorphism $h_1\implies h_2$, that would be too restrictive (for instance it would imply, if we pick $E=D, h_1=h_2 = id_D$, that $D$ has no center, and if we take $S=\emptyset$, it would imply that $C$ has no center, which is a bit sad) - what we are saying is that there is a unique isomorphism appropriately commuting with the data, and here the data is "identities on $C$"; but since $h_1(f(c))$ and $h_2(f(c))$ are not strictly equal, we must have a substitute for this identity : it is the isomorphism we use to identify the two objets and -oh how lucky we are !- we are given such an isomorphism, it's $\theta$ (think of this again with analogy to the categorical product, $h_1$ is the object, $\theta_1$ is the projections).
The claim now follows directly from the definition : indeed a fully faithful functor creates isomorphisms, so if you have your isomorphism $\theta : h_1\circ f \implies h_2\circ f$, then it's of the form $\eta f$, for some unique (by faithfulness again) isomorphism $\eta : h_1\implies h_2$