Equivalent definitions of Saturated multiplicative system in a category

Question

$\DeclareMathOperator{\id}{id}$I'm discovering the localization of categories, subcategories and functors. Amongst the first thing that comes up in this chapter, in the book I'm reading, is the definition of a saturated multiplicative system. For completeness' sake, let me first state the definitions I'm working with. A collection $\mathscr{S}$ of morphisms in a category $\mathscr{C}$ is a right multiplicative system if :

1. For any object $X$ in $\mathscr{C}$, $\id_{X}\in\mathscr{S}$.
2. For any pair $f,g$ of composable morphisms in $\mathscr{S}$, $g\circ f\in\mathscr{S}$
3. Given a morphism $f:X\to Y$ in $\mathscr{C}$ and $s:X\to X'$ in $\mathscr{S}$ there exists $t:Y\to Y'$ in $\mathscr{S}$ and $g:X'\to Y'$ in $\mathscr{C}$ such that $tf=gs$ (i.e. there is an object and two morphisms making a commutative square).
4. Given a pair of parallel morphisms in $\mathscr{C}$ $f,g:X\to Y$, if there exists $s\in \mathscr{S}:W\to X$ such that $fs=gs$ then there exists $t\in\mathscr{S}:Y\to Z$ such that $tf=tg$.

One defines in the same way a left multiplicative system by reversing the arrows, and $\mathscr{S}$ is said to be multiplicative if it is both a left and right multiplicative system. Finally $\mathscr{S}$ is said to be saturated if it satisfies the following axiom :

5. For any sequence of morphisms $X\xrightarrow[]{f}Y\xrightarrow[]{g}Z\xrightarrow[]{h}W$, such that $gf$ and $hg$ are in $\mathscr{S}$, then $f\in \mathscr{S}$.

My goal is to prove that, if $\mathscr{S}$ is a multiplicative system (both left and right) then being saturated is equivalent to

6. For any sequence of morphisms $X\xrightarrow[]{f}Y\xrightarrow[]{g}Z\xrightarrow[]{h}W$, such that $gf$ and $hg$ are in $\mathscr{S}$, then $g\in \mathscr{S}$.

Upon looking up more details on the subject online, I found that 6. was sometimes taken as the definition of saturated.

I've tried to prove $5. \implies 6.$ by naively applying 1,2,3,4 (and their dual counterparts) but to no avail. It seems like it should be a straightforward proof, but anytime I try anything I run into a wall, for instance not being able to show that some composition lies in $\mathscr{S}$ hence not being able to apply 5 to conclude. My instinct was to somehow using 1/2/3/4 I'd be able to produce a morphism $W\to V$ and that this morphism would satisfy the conditions of 5. hence allowing me to conclude but while I was able to define such morphisms, it didn't work.