So in Vakils notes, for an arbitrary scheme $X$ he takes a distinguished base to be the sub category of open sets on $X$, consisting of open affines, where we only remember the inclusion of distinguished opens. He goes on to demonstrate that a sheaf on a distinguished base induces a unique sheaf on $X$, which is unique up to unique isomorphism.
I am wondering if we can extend this idea to two mildly different scenarios. The first, is let $f:X\rightarrow Y$ be a morphism of schemes. Then the stacks project mentions here that the open affines of $X$ which map into open affines of $Y$ form a basis for the topology of $X$. Could we then apply the same trick and only remember the inclusions of distinguished opens of such open affines and define sheafs that way? I think the answer is yes, as if $F$ is the sheaf on such a distinguished base, I think that defining: $$\mathcal{F}(U)=\left\{(s_x)\in \prod_{x\in U}F_x: \forall x\in U, \exists V_{f(x)}\text{ open and affine}, U_x\in B\subset f^{-1}(V_{f(x)}), g\in F(U_x), \text{s.t.} g_{x'}=s_{x'}\forall x'\in U_x\right\}$$ where $B$ is our distinguished base. I believe that it is easy enough to show that this is a sheaf, but I'm not sure how to show this would be unique up to unique isomorphism?
The second case I am much less sure about. Let $f_1:X_1\rightarrow Y$, $f_2:X_2\rightarrow Y$, and $Z=X_1\times_Y X_2$. Then we know there is a cover of $Z$ of open affines of the form $\operatorname{Spec}(B_1\otimes_A B_2)$, where $\operatorname{Spec A}\subset Y$, and $\operatorname{Spec}B_j\subset f_j^{-1}(\operatorname{Spec}A)$. Is there a way to construct sheafs on this base where we only consider open affines of this form, and where the inclusions are only the 'easy' distinguished opens of $\operatorname{Spec}B_1\otimes_AB_2$, i.e. those that correspond to fibre products of distinguished opens? I think this is probably not true, since there exists distinguished opens of $\operatorname{Spec}k[x]\otimes_kk[x]$ which can't be covered by $U_{a_i}\times_{\operatorname{Spec} k}U_{b_i}$.
I am ultimately just trying to get around gluing morphisms of sheaves together, and instead just trying to define morphisms on a distinguished base that commute with restrictions since it seems to me to be a computationally easier route.