Let C be a V-enriched category and 1 be a terminal object of C. V is not necessarily a closed category, and C does not necessarily have an internal hom (nor is C even necessarily a monoidal category).
Is there a term for such an enrichment in which Hom(1,-):C->V is a full functor?
Are there any non-obvious sufficient conditions which imply that Hom(1,-):C->V is full?
Background: in a certain sense, when Hom(1,-):C->V is full, it means that from V's perspective the only way to turn a morphism f:1->A into a morphism 1->B is to find some g:A->B and compose $g\circ f$. This is a consequence of the fact that every V-map C(1,A)->C(1,B) arises as Hom(1,g) for some C-map g. So in this scenario, C "knows about" all the ways of turning a 1->A into a 1->B (or, at least knows about all the ways that V knows about).
I am interested in learning more about the properties of these sorts of enrichments, but specifically without assuming that V is a closed category (which seems to be the case that gets the most attention in the literature I've been able to find).