Why do we use the smash product in the category of based topological spaces? I was telling someone about the smash product and he asked whether it was the categorical product in the category of based spaces and I immediately said yes, but after a moment we realized that that wasn't right.  Rather, the categorical product of $(X,x_0)$ and $(Y,y_0)$ is just $(X\times Y,(x_0,y_0))$.  (It seems like in any concrete category $(\mathcal{C},U)$, if we have a product (does a concrete category always have products?) then it must be that $U(X\times Y)=U(X)\times U(Y)$.  But I couldn't prove it.  I should learn category theory.  Maybe functors commute with products or something.)  Anyways, here's what I'm wondering: is the main reason that we like the smash product just that it gives the right exponential law?  It's easy to see that the product $\times$ I gave above has $F(X\times Y,Z)\not\cong F(X,F(Y,Z))$ just by taking e.g. $X=Y=Z=S^0$.

This is pretty much (derived from, I guess) Jonas Meyers answer, but a bit more concrete, and as far as I know why we're interested in it. There is an adjunction $\hom_*(\Sigma X, Y)\cong\hom_*(X,\Omega X)$, where $\Sigma X:=S^1\wedge X$ and $\Omega X:=\hom_*(S^1,X)$. If we define $\pi_n(X):=\pi_0(\Omega^n X)$, or indeed $\pi_n(X):=[S^n,X]_*$, we get $\pi_n(X):=\pi_0(\Omega^n X)\cong[S^0,\Omega^n X]_*\cong[\Sigma^n S^0,X]_*\cong[S^n,X]_*$, which is an interesting relationship.
