TITLE: Why is the smash product in a certain slice category associative?
QUESTION [6 upvotes]: I am trying to show the following. Let $(V, \times , \ast)$ be a cartesian closed symmetric monoidal category (so the monoidal product is the ordinary product). Consider the category $V_\ast$ of based objets, objets under $\ast$, the terminal objet of $V$. One then defines the smash produt in $V_\ast$ of v and w to be the pushout
$$\require{AMScd} \begin{CD}
v \amalg w @>>> v \times w ;\\
@VVV @VVV \\
\ast @>>> v \wedge w.
\end{CD}$$
Appearently, this is supposed to follow because $V$ is cartesian closed and by this, $v \times -$ preserves colimits. But I don't see how this gives my proposition, unfortunately.
Anyhelp would be appreciated.

REPLY [3 votes]: Zhen writes: "I suppose it may be possible to verify the associativity of ∧ using only the fact that × preserves colimits in each variable separately, but this looks considerably more tedious than either approach outlined above."
Actually it is straight forward: $U \wedge (V \wedge W)$ is the quotient of $U \times (V \wedge W)$ which identifies $* \times (V \wedge W)$ with $U \times * \times *$. But $U \times (V \wedge W)$ is the quotient of $U \times V \times W$ which identifies $U \times * \times W$ with $U \times V \times *$ (in particular $U \times * \times *$). It follows that $U \wedge (V \wedge W)$ is the quotient of $U \times V \times W$ which identifies $* \times V \times W$ with $U \times V \times *$ and $U \times * \times W$. The reasoning applies to $(U \wedge V) \wedge W$. In fact, this shows that both are isomorphic to the object $U \wedge V \wedge W$ which classifies tripointed morphisms.
By the way, this whole construction can be seen as a special case of a more general theory about monoidal monads (see for example arXiv/1205.0101) applied to the monoidal monad $X \mapsto * + X$.
Finally let me mention that we cannot apply this to $(\mathsf{Top},\times)$, since here $\times$ doesn't preserve colimits in each variable. In fact, the smash product of pointed spaces is not associative in general (proof?).