= Solution
On the <principal open subscheme> $D(a)\subseteq\operatorname{Spec}A$, the induced <morphism of sheaves> is the <ring homomorphism>
$$
A_a\longrightarrow B_{\alpha(a)},\qquad \frac{b}{a^n}\longmapsto\frac{\alpha(b)}{\alpha(a)^n}.
$$
If $\alpha$ is <injective>, this map is injective by <exactness of localization>: it is the localization of the injective $A$-module map $A\to B$. Explicitly, if the displayed fraction vanishes, some power of $\alpha(a)$ annihilates $\alpha(b)$; injectivity gives $a^mb=0$, so the original fraction also vanishes. The principal opens form an <open basis>, so the <kernel sheaf> is zero on a basis and hence zero everywhere. Conversely, an injective <morphism of sheaves> induces an injection on <global sections>. On the whole <affine scheme> these sections are $A\to B$, exactly $\alpha$. Thus
$$
\boxed{\alpha\text{ is injective}\ \Longleftrightarrow\ \phi\text{ is injective}.}
$$
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