multiplication in H-space and loopspace of the H-space Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy.
Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $ \circ $ on the loopspace be given by concatenating loops.
We also have a multiplication on the loopspace given by pointwise multiplication of loops.  Since the loops are based at the identity this also give a loop based at the identity. Call this multiplication $\mu$.
Are the multiplication maps $\mu$, $\circ$ homotopic?
I am trying to verify the assertion in Odd primary exponents of Moore Spaces that if the power map $\dot k: X \to X$ sending $x$ to a choice  of the $k$th power $x^k=(x(x(x(x...))$ $k$ times,  is null homotopic, then the map $k^\circ$ sending $\gamma \in \Omega X$ to its kth homotopy exponent is also null homotopic.
Define $k^\mu$ to be the $k$-th power map on the loopspace using pointwise multiplication of loops.
Now if $\mu$ and $\circ$ are indeed homotopic, then since $k^\mu$ is null homotopic, it follows that  $k^\circ$ is also null homotopic.  
So is it true that $\mu$ and $\circ$ are homotopic?

This follows by an Eckmann-Hilton argument and the Yoneda lemma.  For any pointed space $Y$, $\mu$ and $\circ$ each give a binary operation on the set $[Y,\Omega X]$ of homotopy classes of pointed maps $Y\to\Omega X$.  By Yoneda, $\mu$ and $\circ$ are homotopic iff these binary operations are equal for all $Y$.
Now observe that for any $Y$, the two binary operations on $[Y,\Omega X]$ both have the constant map as an identity, and they satisfy the interchange law $\mu(a\circ b,c\circ d)=\mu(a,c)\circ\mu(b,d)$ (since this equation holds on the nose for points of $\Omega X$, as can easily be verified by plugging in the definitions of $\mu$ and $\circ$).  By Eckmann-Hilton, they must be equal.  Explicitly, for any $a,b\in [Y,\Omega X]$ and $1\in[Y,\Omega X]$ the constant map, we have $$\mu(a,b)=\mu(a\circ 1,1\circ b)=\mu(a,1)\circ\mu(1,b)=a\circ b.$$
