Let $X$ be a topological space and $X = \cup U_i$ an open cover of $X$. Suppose we have sheaves $F_i$ on $U_i$ along with isomorphisms $\phi_{ij}: F_i |_{U_i \cap U_j} \rightarrow F_j |_{U_i \cap U_j}$ that agree on triple overlaps. Then we know that we can glue these $F_i$'s to form a sheaf on $X$ such that $F|_{U_i}$ is isomorphic to $F_i$ and this choice of $F$ is unique up to unique isomorphisms. (This is the exact statement of the exercise I did)
I have a small question regarding the "unique up to unique isomorphism" bit. I was wondering if I specify my $F$ to satisfy $F|_{U_i} = F_i$ (in contrast to isomorphism as above), then is the choice for $F$ unique?
In general it would be impossible to arrange your $F$ to have the property $F|_{U_i}=F_i$ for all $i$, precisely because the $U_i$ may overlap and the $F_i$ may have nothing to do with each other on a set-theoretic level.