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 $A_3$.