A relative category is a category $C$ with a subcategory $W$ containing all the objects of $C$.
Given a relative category $(C,W)$, $W$ is said to satisfy the ``2 implies 6'' property if, for any collection of three composable maps,
$$X\rightarrow Y\rightarrow Z\rightarrow A$$
the presence of the composites $X\rightarrow Z$ and $Y\rightarrow A$ in $W$ implies that each individual map is in $W$ (and so also the triple composition).
The property I'm more familiar with from thinking about weak equivalences is the ``2 implies 3" property, which says that, given a pair of composable maps
$$X\rightarrow Y\rightarrow Z$$
the presence of any two of the maps
$$X\rightarrow Y$$ $$Y\rightarrow Z$$ $$X\rightarrow Z$$
in $W$ implies that the third is as well.
The "2 implies 6" property implies the "2 implies 3" property, and I've been told that "2 implies 6" is a strictly stronger property.
QUESTION: What is the basic example of a relative category $(C,W)$ where $W$ satisfies "2 implies 3", but not "2 implies 6"?
Edit: By "basic", I mean what is an example which comes up in applications, or better yet, what is the example to keep in mind?