algebraic reason that homs to affine schemes factor uniquely through open affine subschemes

I notice that an affine open subscheme $X = operatorname{Spec} R$ of an affine scheme $Y = operatorname{Spec} S$ has the property that any ring hom $Srightarrow T$ that factors through $Srightarrow R rightarrow T$ does so uniquely (that is the hom from $R$ to $T$ is unique). In case $X$ is distinguished it is clear algebraically why this is the case: the image of $S$ in $R$ somehow generates $R$ if you allow inverses. The same is true if $X$ is not necessarily distinguished but $S$ is integral. But in general, what is the algebraic reason for this uniqueness? Is it generally true that, in the aforementioned sense the image of $S$ in $R$ somehow generates $R$?