Let,$(X, \mathcal O_X)$ be a scheme and $\mathcal B$ be a quasicoherent sheaf of $\mathcal O_X$ algebra.Then for any $U =spec(A)$ affine open of $X$ the map $\mathcal O_X(U) \to \mathcal B(U)$ induces a morphism of schemes given by $\pi_U: Spec(\mathcal B(U)) \to U$
I want to prove that it's an affine morphism
My attempt: Let's cover $U =SpecA$ by affine opens $D(f) =SpecA_f$.Let's denote the image of an arbitrarily fixed $f$ as $\tilde f$ under the map $\mathcal O_X(U) \to \mathcal B(U)$.Then we know that inverse image of each such $D(f)$ under $\pi_U$ will be $\mathcal (B(SpecA))_\tilde{f}$,which is affine as $\mathcal B$ is quasicoherent.Thus we are done.
Doubt: It seems from this we can conclude any scheme morphism from $SpecR$ to $SpecS$ is also affine which is not true.
Please correct me if the proof is wrong.
Any help from anyone is welcome.