Let $\mathcal{C}, \mathcal{D}, \mathcal{E}$ be (symmetric?) monoidal categories, and $H : \mathcal{C} \times \mathcal{D} \to \mathcal{E}$ be a functor that is monoidal in both arguments, ie. $H(C,-)$ and $H(-,D)$ are (strong) monoidal for all objects $C$ and $D$. And take two monoids $M : \Delta \to \mathcal{C}$ and $N : \Delta \to \mathcal{D}$ (where $\Delta$ is the free monoidal category on a monoid, 1).

$H(M 1, N 1)$ is a monoid in two different ways, as $H(M1, -)$ and $H(-,N1)$ both preserve monoids.

I'd like to use an Eckmann-Hilton style argument to prove that the two monoids coincide and are commutative.

Do I need additional assumptions to prove this is true? The concrete example I had in mind is: $\mathcal{C} = h\mathrm{Top}_*^{op}$, $\mathcal{D} = h\mathrm{Top}_*$, $\mathcal{E} = \mathrm{Set}$ and $H = \mathrm{Hom}$, $M$ is the "cogroup" on $S^1$ used to define the fundamental group and $N$ is some topological group and I'd like to conclude that $\pi_1(N,id_N) = \mathrm{Hom}_{h\mathrm{Top}_*}((S^1,1),(N,id_N))$ is abelian.

Also, is there a category $\mathcal{K}$ representing the functors like $H$, ie. $$\mathrm{Hom}_\mathrm{MonCat}(\mathcal{K},\mathcal{E}) \cong \{ H : \mathcal{C} \times \mathcal{D} \to \mathcal{E} \text{ st. } H(C,-) \text{ and } H(-,D) \text{ are monoidal for all } C, D \}$$ sort of like a tensor product? Is it of any use?