Name of some category with two objects Does the category that consists of two objects and exactly one non-identity morphism that connects both objects have a specific name?
 A: The interval category. It is often denoted $\mathbf{I}$. Note that if $\mathbf{C}$ is category, then $\mathbf{C}^\mathbf{I}$ is the arrow category of $\mathbf{C}$.
A: As goblin suggests, "the interval category" is good, and it has the imprimatur of the nlab. I might also call it "the arrow category". If I say "the interval category", I somehow feel the need to clarify that I'm talking about the category with one nontrivial arrow, rather than one nontrivial isomorphism. I might also use the notation $\uparrow$, although this is hard to pronounce. 
Another possibility is "the walking arrow" (as opposed to "the walking isomorphism"). In general, if you have a functor $F$ which is isomorphic to $\mathrm{Hom}(X,-)$, then you might call $X$ "the walking $F$" (in this case we have an isomorphism of 2-functors $\mathrm{Arr} \cong \mathrm{Hom}(\uparrow, - ): \mathsf{Cat} \to \mathsf{Cat}$ where $\mathrm{Arr}$ sends a category $C$ to its arrow category $\mathrm{Arr} C$). This terminology probably feels a little too informal in some contexts, although it's fairly precise.
I suppose you could also say "the universal arrow" (as opposed to "the universal isomorphism").
A: It might also be denoted $\mathbf 2$ as it is the category associated to the cardinal poset $\mathbf 2 = \{0<1\}$.
