Let $X_i$ be schemes $(i \in I)$, $X_{ij} \subseteq X_i$ open subschemes with $X_{ii} = X_i$ along with isomorphisms $f_{ij}:X_{ij} \rightarrow X_{ji}$ such that $f_{ik} |_{X_{ij} \cap X_{ik}} =f_{jk} |_{X_{ji} \cap X_{jk}} f_{ij} |_{X_{ij} \cap X_{ik}}$. We know that we can glue these schemes together to obtain a unique scheme up to unique isomorphism.
I am confused about the following. Here when they say unique up to unique isomorphisms, I am guessing that it means it is unique up to unique isomorphism satisfying certain relations with the given $f_{ij}$'s. But I wasn't quite sure and I was hoping someone could clarify what it means by unique up to unique isomorphisms here. Thank you!
As a hint, I'd suggest first determining what, exactly, is the connection between the glued scheme $X$ and the various $X_i$'s and maps $f_{ij}$. As you say, this is the important property. For example, there are various open inclusions $f_i : X_i \hookrightarrow X$, no doubt commuting with the maps $f_{ij}$.
Then the universal property is that if $X'$ is any other scheme having the same diagram of maps from the $X_i$'s and $X_{ij}$'s, there exists a unique isomorphism $X \to X'$ that commutes with the diagrams. (In other words, $X$ should be the 'initial object' in the category of "objects having the appropriate collection of morphisms from the $X_{ij}$'s".)