I'm reading through Lurie's Higher Topos Theory and I'm not conviced by the proof of corollary 2.3.2.4. it asserts that it $i:A\rightarrow A'$ is inner anodyne and $j:B\rightarrow B'$ is a cofibration, then $$(A\times B')\coprod\_{A\times B}(A'\times B)\rightarrow A'\times B'$$
is inner anodyne.
How does the fact that the class of maps $A\_3$ is stable under smash products with cofibrations can be extended to a statement about all inner anodyne maps?



A smash product of an inner anodyne map with a cofibration is inner anodyne
Just to make it clear: in this context "smash product" of $i$ and $j$ means the map 
$$
i\square j \colon (A\times B')\amalg\_{A\times B} (A'\times B)\to A'\times B'
$$
constructed from $i\colon A\to A'$ and $j\colon B\to B'$. (I can't find a place in the book where Jacob defines "smash product" properly, though this is how he uses it.)


The first observation is that "smash product" is associative:
$$
(i\square j)\square k \approx i\square (j\square k).
$$
The second observation is that it is commutative: $i\square j\approx j\square i$.


The third observation is that the smash product of two monomorphisms is a monomorphism (=cofibration).


Fourth observation: if a "weakly saturated" class $C$ of maps (such as inner anodynes) is generated by a class $G$, then to show $C\square j\subseteq C$ it suffices to show $G\square j\subseteq C$.


To show that if $i$ is inner anodyne and $j$ a monomorphism, then $i\square j$ is inner anodyne, it is enough to show it for $i$ in a generating class for inner anodynes, e.g., $i$ in $A\_3$. So if $i=i'\square u$ with $i'$ a monomorphism and $u\colon \Lambda^1\_1\to \Delta^2$, we get
$$
i\square j = (i'\square u)\square j \approx (i'\square j)\square u,
$$
which is in $A\_3$. 


**Added.** About the fourth observation: given a class of maps $S$, write $\overline{S}$ for the weak saturation of $S$, i.e., the class containing $S$ and closed under the various constructions that weak saturated classes are supposed to be closed under. Then in general, we have that
$$ \overline{S}\square j \subseteq \overline{S\square j}.$$


You can prove this directly from the definition of weak saturated class. A different argument uses the idea that $\overline{S\square j}$ is exactly the class of maps which have the left-lifting-property with respect to the class of maps with the right-lifting-property with respect to $S\square j$ (this uses the small-object argument). I don't feel like writing it up here: see for instance section 15 in my notes on quasicategories: <http://www.math.uiuc.edu/~rezk/quasicats.pdf>


