Multiplicative system in a category

Let $\mathcal{C}$ be a locally small category, using the definable-class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{C}\mathbf{A}\mathbf{T}$ is not formed. A two-sided multiplicative system $S$ is a class of arrows satisfying:

1. Every $1_X$ belongs to $S$, and composites of composable members belong to $S$.
2. Given $f:U\to Y$ and $t:V\to Y$ in $S$, there exist $a:W\to U$ in $S$ and $b:W\to V$ with $fa=tb$. Dually, given $f:X\to U$ and $s:X\to V$ in $S$, there exist $a:U\to W$ in $S$ and $b:V\to W$ with $af=bs$.
3. For parallel $f,g:X\to Y$, existence of $t:Y\to Z$ in $S$ with $tf=tg$ is equivalent to existence of $s:W\to X$ in $S$ with $fs=gs$.

For the locally small localization construction we additionally require either that $\mathcal{C}$ is small or that, for each $X$, a set $S_X$ of denominators into $X$ is supplied such that every $s:U\to X$ in $S$ admits $a:V\to U$ with $sa\in S_X$. The map $a$ need not lie in $S$. These size data are separate from the fraction axioms; local smallness of $\mathcal{C}$ alone is insufficient. Objects and arrows have the types prescribed in Category, object, morphism, domain, codomain, identity, composition, and hom-collection.