I'd like to go on discussing the proof which I started to discuss here.
The book says sending $(s_x)_x\in\tilde{\mathscr{F}}(U)$ to $(\varphi_x(s_x))_x\in\tilde{\mathscr{G}}(U)$ defines a morphism $\tilde{\varphi}:\tilde{\mathscr{F}}\to\tilde{\mathscr{G}}$. By Prop. 2.23(3) this is the unique morphism making the diagram (2.7.1) commutative.
The diagram says that we have $\iota_\mathscr{G}\circ\varphi=\tilde{\varphi}\circ\iota_\mathscr{F}$.
Question 1: To verify this assertion I first checked that the so defined $\tilde{\varphi}$ is really a morphism. Furthermore I saw that defining $\tilde{\varphi}$ in such a way makes the diagram commutative. In fact we need to define $\tilde{\varphi}$ in exactly that way for the images of $\iota_\mathscr{F}$. So for uniqueness it's only left to show that there is no other choice for the elements of $\tilde{\mathscr{F}}(U)$ which are no images of $\iota_\mathscr{F}$. Here I don't see how Prop. 2.23(3) would help us. That propositions states (translated to our problem) that $\tilde{\varphi}$ is already defined by $\tilde{\varphi}_x$. So how do we see that the maps $\tilde{\varphi}_x$ are uniquely defined?
Question 2: In the end of the proof we read that the uniqueness of $(\tilde{\mathscr{F}},\iota_\mathscr{F})$ is a formal consequence. What is meant by that?