Let us call a group object $G$ in a category $\mathcal C$ rigid, if it has the following property: For every group object $X$ in $\mathcal C$, every morphism $G\to X$ in $\mathcal C$ respecting the unit sections is already a morphism of group objects.
As a formal consequence, rigid group objects are commutative (because the inversion is a group object morphism).
This definition is motivated by the following fact: If $\mathcal C$ is the category of algebraic varieties over $\mathbb C$ (say), then the rigid group objects in $\mathcal C$ are precisely the complex abelian varieties. Indeed, that abelian varieties enjoy the rigidity property is a "standard fact", and the converse follows quickly from Chevalley's structure theorem (every algebraic group is an extension of an abelian variety by an affine group).
If $\mathcal C$ is the category of sets or of hausdorff topological spaces, there are no interesting rigid group objects. I am puzzled with the case where $\mathcal C$ is the category whose objects are the topological spaces which are homotopy equivalent to CW-complexes and morphisms are continuous maps up to homotopy. A group object in this category is commonly called $H$--group.
Is it true that the rigid $H$--groups are exactly the products of circles?