Extending a morphism on stalks by base change

This question is from Qing Liu Algebraic Geometry and Arithmetic Curves chapter 3 exercise 2.5 on pg. 96. I’ve approached this is the same spirit as the answer given to this highly related question.

Let $S$ be a locally Noetherian scheme, $X$, $Y$ be $S$-schemes of finite type. Fix $sin S$, and let $varphi:Xtimes_Soperatorname{Spec}mathcal{O}_{S,s}to Ytimes_Soperatorname{Spec}mathcal{O}_{S,s}$ be an $S$-morphism of schemes. Show that there exists an open set $Uni s$ and a morphism of $S$-schemes $f:Xtimes_SUto Ytimes_SU$ such that $varphi$ is obtained from $f$ by the base change $operatorname{Spec}mathcal{O}_{S,s}to U$. If $varphi$ is an isomorphism, show that there exists such an $f$ which is moreover an isomorphism.

As a first step, I’d like to argue that this problem is affine in $S$, $X$, and $Y$. For $S$ this is trivial, as we may take $U$ to be an open affine. For $X$ and $Y$, I’d like to argue along the following lines: if for every open affine $Vsubset Y$, and every open affine $Wsubset X$ for which $varphi(Wtimes_Soperatorname{Spec}mathcal{O}_{S,s})subset Vtimes_Soperatorname{Spec}mathcal{O}_{S,s}$, there is an $f_W$ extending $varphi|_{Wtimes_Soperatorname{Spec}mathcal{O}_{S,s}}$ by base change, then the $f_W$ agree on overlaps and glue to a morphism $f$ extending $varphi$.

Question 1: Is this argument correct, and how do I make it more formal?

Reducing to the affine case, I may assume $S=operatorname{Spec}R$ with $R$ Noetherian (since $S$ is locally Noetherian), and that $X = operatorname{Spec} B$ and $Y = operatorname{Spec} A$ for finite-type $R$-algebras $A$ and $B$. Then if $mathfrak{p}$ is the prime corresponding to $sin S$, $varphi^#(Y):Aotimes_R R_mathfrak{p}to Botimes_RR_mathfrak{p}$, so $varphi$ extends to a neighborhood of $s$ if and only if there is some $hin R-mathfrak{p}$ such that there is a homomorphism
$$varphi^#(Y)otimes pi:Aotimes_R R_hto Botimes_R R_h$$
where $pi:R_hto R_mathfrak{p}$ is the canonical map.

To find such an $h$, write $A = R_mathfrak{p}[T_1,dots,T_n]/(f_1,dots,f_r)$ and $B=R_mathfrak{p}[S_1,dots,S_m]/(g_1,dots,g_s)$ where the $f_iin R[T_1,dots,T_n]$ and $g_jin R[S_1,dots,S_m]$. Define $$Phi:R_mathfrak{p}[T_1,dots,T_n]to R_mathfrak{p}[S_1,dots,S_m]/(g_1,dots,g_s)$$
where $Phi(p(T))$ is a coset representative of $varphi(p(T)+(f_1,dots,f_r)$. Then $Phi(T_i) = q_i(S)in R_mathfrak{p}[S_1,dots,S_m]$, and $Phi(f_j) in (g_1,dots,g_s)$, so
$$Phi(f_j) = sum_{k=1}^s frac{a_{jk}}{s_{jk}}g_k.$$
where the $a_{jk}in R$ and $s_{jk}in R-mathfrak{p}$
and
$$Phi(T_i) = sum_{k_igeq 0}frac{a_{k_1,dots,k_m,i}}{s_{k_1,dots,d_m,i}}S^{k_1}cdots S^{k_m}$$
where $a_{k_1,dots,k_m,i}in R$ and $s_{k_1,dots,k_m,i}in R-mathfrak{p}$. I’m tempted to define $h$ to be the product of all such $s$ and argue that $Phiotimes pi$ is well-defined as a mapping $Aotimes_RR_hto Botimes_RR_h$, but the notation has become extremely cumbersome.

Question 2: Am I on the right track/is there a more elegant approach?