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?



(Homotopy theory) When does the 2 of 3 property not imply 2 of 6?
I'm not sure that any examples naturally come up, of cases where you have the 2 out of 3 condition but not the 2 out of 6. Of course, if membership in W is defined by requiring certain functors to take a morphism to isomorphisms (as is so often the case in applications), then you always have 2 out of 6 (because a morphism that has both a left inverse and a right inverse always has an inverse). 


In Quillen's model category axioms 2 out of 3 is an axiom and 2 out of 6 follows from this and the other axioms.


