at.algebraic topology - homotopy associative $H$-space and $coH$-space

Let $[X, Y]_0$ denote base point preserving homotopy classes of maps $Xrightarrow Y$. A multiplication on a pointed space $Y$ is a map $phi: Ytimes Yrightarrow Y.$ From this map, we can define a continuous map for each pointed space $X$, $phi_X: [X, Y]_0times [X, Y]_0rightarrow [X, Y]_0,$
by the composition $$phi_X (alpha, beta)(x)=phi(alpha(x), beta(x)).$$
If $([X, Y]_0, phi_X)$ is a group for each $X$, then $(Y, phi)$ is called a homotopy associative $H$-space.

A $coH$-space is defined from a comultiplication, namely, a map $psi: Xrightarrow Xvee X.$ Then, for each pointed space $Y$, we can define a function $psi^Y: [X, Y]_0times [X, Y]_0rightarrow [X, Y]_0$ in this way:
$$psi^Y(alpha, beta)=(alphaveebeta)circpsi.$$ If $([X, Y]_0, psi^Y)$ is a group for each $Y$, then $(X, psi)$ is called a homotopy associative $coH$-space.

So, as we can see, if we have a homotopy associative $coH$-space $(X, psi)$ and a homotopy associative $H$-space $(Y, phi)$, then we can define two group structures on the space $[X, Y]_0$. My question is: are they "equivalent" in some sense? Obviously, whatever $phi$ or $psi$ is, the zero element of the group is the constant map in $[X, Y]_0.$ However, the two group structures do depend on the choice of $phi$ and $psi$, which seems have little relationship with each other.