# “discrete” objects of a $2$-category

Let $$\mathcal{K}$$ be a $$2$$-category. Is there a special name of those objects $$B \in \mathcal{K}$$ which have the property that the category $$\mathrm{Hom}_{\mathcal{K}}(B,C)$$ is essentially discrete for all $$C \in \mathcal{K}$$? This means that for every two morphisms $$f,g : B \to C$$ any $$2$$-morphism $$f \to g$$ is an isomorphism, and it is unique if it exists.

Also, is there some intuition what this property actually means, intuitively?

Assuming that $$\mathcal{K}$$ has more properties (such as the existence of certain limits), can we simplify the property perhaps to some "internal" information of $$B$$?

Here is the example which motivates my question: Consider the $$2$$-category $$\mathrm{Cat}_{c\otimes/R}$$ of cocomplete $$R$$-linear tensor categories and let $$A$$ be a commutative $$R$$-algebra. Then $$\mathrm{Mod}(A) \in \mathrm{Cat}_{c\otimes/R}$$ has this property, since $$\mathrm{Hom}_{c\otimes/R}(\mathrm{Mod}(A),\mathcal{C}) \simeq \mathrm{Hom}_R(A,\mathrm{End}(1_\mathcal{C}))$$ and the latter is a set. I have recently proven that $$\mathrm{Qcoh}(X)$$ has this property as well, where $$X$$ is any quasi-compact quasi-separated $$R$$-scheme. When $$X$$ is an algebraic stack, usually $$\mathrm{Qcoh}(X)$$ doesn't have this property.

I would call those objects "codiscrete" or "co-0-truncated", since "discrete" and "0-truncated" are used for the dual property, e.g. here and here. It's equivalent to saying that $$B$$ is equivalent to the copower $$B \odot (\cdot \rightrightarrows \cdot)$$.