ScalingStacks

Let $E$ denote, as in §6, the discrete nerve of $I[1]$. We define a categorical homotopy between maps $f,g\colon U\rightrightarrows V$ of Segal spaces to be any one of the following equivalent data: a map $H\colon U\times E\rightarrow V$, a map $H^{\prime}\colon U\rightarrow V^{E}$, or a map $H^{\prime\prime}\colon E\rightarrow V^{U}$, making the appropriate diagram commute:

If $U$ and $V$ are discrete nerves of categories $C$ and $D$, then the categorical homotopies of maps between $U$ and $V$ correspond exactly to natural isomorphisms of functors between $C$ and $D$.