In Vistoli's FGA explained, when he introduces fibered categories he defines Cartesian arrows and pullbacks. The definition of Cartesian arrow makes sense and is standard, and then he defines pullback:
If $\xi\rightarrow\eta$ is a Cartesian arrow of $F$ mapping to an arrow $U\rightarrow V$ of $C$, we also say that $\xi$ is a pullback of $\eta$ to $U$.
He goes on to remark that, given two pullbacks $\phi:\xi\rightarrow\eta$ and $\phi':\xi'\rightarrow\eta$ of $\eta$ to $U$, then $\xi$ and $\xi'$ are uniquely isomorphic.
This gives me pause because I feel that if $\phi$ and $\phi'$ induce different morphisms $U\rightarrow V$, then I don't see why this should be true. Indeed, looking back at the definition of pullback, it seems essential that the data of the specific arrow $U\rightarrow V$ needs to be specified in order to make a statement like "the pullback is unique up to unique isomorphism." This is needed for other examples of pullbacks in math anyway. So I just want confirmation that indeed this is the case, because the text of the definition is at best unclear to me.