Does forgetful functor $U:\mathbf{Sch}\to \mathbf{Top}$ preserves fibre products?

Question

Let $\mathbf{Sch}$ be a category of schemes and let $\mathbf{Top}$ be a category of topological space. We have a forgetful functor $U:\mathbf{Sch}\to \mathbf{Top}$. Does this functor preserves fibre products? I don't know whether this is true for affine schemes - for any ring $R$ and $R$-algebras $f:R\to A$ and $g:R\to B$, can we identify $\mathrm{Spec}(A\otimes_{R}B)$ with $\{(\mathfrak{p}, \mathfrak{q})\in \mathrm{Spec}(A)\times \mathrm{Spec}(B)\,:\,f^{-1}(\mathfrak{p}) = g^{-1}(\mathfrak{q})\} $?