Remark 7.6.5.3. There is a tautological epimorphism of simplicial sets
\begin{eqnarray*} ( \bullet \rightrightarrows \bullet ) & = & \Delta ^1 {\coprod }_{ \operatorname{\partial \Delta }^1 } \Delta ^1 \\ & \twoheadrightarrow & \Delta ^1 {\coprod }_{ \operatorname{\partial \Delta }^1 } \Delta ^0 \\ & = & \Delta ^1 / \operatorname{\partial \Delta }^1. \end{eqnarray*}
It follows from Example 6.3.4.4 that this epimorphism exhibits $\Delta ^1 / \operatorname{\partial \Delta }^1$ as a localization of $( \bullet \rightrightarrows \bullet )$. In particular, it is both left and right cofinal (Proposition 7.2.1.10).