Remark 7.6.4.2. The simplicial set $( \bullet \rightrightarrows \bullet )$ of Notation 7.6.4.1 is isomorphic to the nerve of its homotopy category $\operatorname{\mathcal{J}}$, which can be described concretely as follows:

The category $\operatorname{\mathcal{J}}$ has exactly two objects $Y$ and $X$.

There are exactly two non-identity morphisms in $\operatorname{\mathcal{J}}$, both of which have source $Y$ and target $X$.