$X$ and $Y$ are schemes(Note they are not necessarily affine), and $(U_i)_{i∈I}$ is an affine open cover of $X$, and let $f_i: U_i → Y$ be morphisms which agree on the overlaps: for all $j,k ∈ I$, we have $f_j|_{U_j∩U_k} = f_k|_{U_j∩U_k}$. Prove there exists a unique morphism $f : X → Y$ such that for all $i ∈ I$, we have $f|_{U_i} = f_i$.
The following are the definitions we use here:
Definition of scheme: A functor $X$ is a scheme if it is (1)local;(2)has an open affine cover.
Definition of local:
(Both of these two definitions are taken in Jantzen's text.)
Many thanks for any help. I have tried on this exercise from this morning almost without any progress(What I have tried so far was searching for a way to use the fact that the $U_i$'s are affine). So pointing out some useful key fact or just give an answer is very ideal.