Fully faithful nerve ⟹ injective on objects?

Question

If $i: C ⟶ D$ (where $C$ is a small subcategory of $D$) is the inclusion functor, and $N_i := {\rm Hom}(i(=), -)$ is the nerve of $i$, is it true that $N_i$ is injective on objects as soon as it is fully faithful? (because if it is fully faithful, $i$ is dense, so I venture that this may be enough to deduce the injectivity of $N_i$ on objects)

Answer 1

It's an odd question, because this depends on your conventions about homsets. Under many conventions, $N_i$ is injective on objects as soon as $C$ admits maps into every object of $D$, the point being that $Hom(c,d)$ is never equal to $Hom(c',d')$, unless perhaps they're both empty. Many foundations assume this disjointness of homsets. But it's awkward to think about equality of sets, and it's hard to see what this would do for you. In any case, this has nothing to do with density of $i$, and you could make your conventions so that $N_i$ needn't be injective on objects even with $i$ an identity if you really wanted: just take a category with two isomorphic objects such that all four morphisms are equal.