Category of groups with homologisms

Definition
Suppose $$\mathcal{V}$$ is a subvariety of the variety of groups. The category of groups with $$\mathcal{V}$$-homologisms is defined as follows:

Note that this is an unconventional category structure on groups. It is definitely not the default category structure. When people talk of the category of groups without mentioning what they mean by the morphisms, they typically do not mean this category and instead mean the usual category of groups where the morphisms are the usual homomorphisms of groups.

Important functors

 * There is a natural functor to this category from the category of groups with marginal homomorphisms.