In a category, a lot of structure can be encoded in the set of arrows that point away from an object. For example, if I let the objects be tensors on some vector space and the Morphisms actions of a Lie Group on that vector space under which the tensors transform, then all the arrows that point away from a tensor encode the information of the whole Lie Group (e.g. all possible ways in which a tensor could be transformed).
However, the "arrows that point away from an object" can not necessarily be comprehended as the elements of a group, it could be elements of some other structure or they could not admit any structure. Therefore I am looking for a category theoretical definition that can help to capture what structure is encoded in the arrows pointing away from an object.
Furthermore, such a notion shall be suitable, if possible, to compare structure of away-pointing arrows of objects of different categories, i.e. to check whether two such sets are equivalent in a structure preserving sense. For example if I would construct a functor from the above mentioned tensor category into another category, I would like to be able to check whether the group structure of away-pointing arrows can be preserved or not.
Any ideas or references to a correlated notion would be helpful. Thanks.