limits of performance

I am trying to understand a topic on limits of performance.

So there is a delay of tau between input and output so the transfer function becomes G(s)*e^-(tau *s).

I am unable to understand the statement "The phase contribution of delay is negative and at crossover frequency is -(tau * omega_c) ,where omega_c is crossover frequency" I don't understand how the phase contribution due to this delay is -(tau*omega_c)?

$e^{-\tau s} \rightarrow e^{-j\omega \tau}=cos(\omega \tau)-jsin(\omega \tau) = 1\angle( -\omega \tau)$
This means unity gain, and a phase lag proportional to angular frequency. At the cut-off frequency of the other the component of the TF, the phase angle contributed by the delay will be found by setting $\omega =\omega_c$, thus: $\phi _{delay}=-\omega_c \tau$ rad.