This is lemma 4.25 of Vistoli's note

Let $S$ be a scheme, $\mathscr{F} \to \mathscr{S}ch/S$ a fibred category. Then $\mathscr{F}$ is a stack over the fpqc site on $S$ iff
(1) $\mathscr{F}$ is a stack over the Zariski site on $S$, and
(2) For every fpqc morphism $V \to U$ over $S$, with $U,V$ affine, $\mathscr{F}(U) \to \mathscr{F}(V \to U)$ is equivalent.

I'm trouble in the last line at step5.

For the notation, see the pdf.

Let $f : V \to U$ be an fpqc morphism over $S$, and $(\eta, \phi) \in\mathscr{F}(V \to U)$. Then by Zariski descent and the step 4, we have $\xi \in \mathscr{F}(U)$ and an isomorphism $\beta : f^* \xi \cong \eta$ in $\mathscr{F}(V)$.
Is this morphism in actually a morphism of descent data?
i.e., is this diagram $\require{AMScd}$ \begin{CD} p_2^* f^* \xi @>{p_2^* \beta}>> p_2^* \eta\\ @V{=}VV @V{\phi}VV\\ p_1^*f^*\xi @>{p_1^* \beta}>> p_1^* \eta \end{CD}

commutative? (where $p_i : V \times_U V \to V$ is the $i$-th projection.)

By the construction of $\beta$, this diagram commutes on $V_i \times_{U_i} V_i = V_i \times_U V_i$. To show this on the whole of $V \times_U V$, by Zariski descent, we must show the commutativity on $V_i \times_U V_j$ for distinct $i,j$. But I can't.

As in step 4, if $\mathscr{F}(U_i \cup U_j) \to \mathscr{F}(V_i \cup V_j \to U_i \cup U_j)$ is equivalence, then I think the diagram is commutative. So I reduce this to the case of quasi-compact $U$.

In stack project, the aouthor ommit this important part.
And in Olsson's "Algebraic spaces and stacks", the author uses (essentially) same argument, but don't mention this commutativity.
And in Lei Fu's "etale cohomology theory", the author leaves so many parts (the whole of the lemma 4.25 of Vistoli's note) as an exercise...

Any help will be appreciated!