Let $X$ be a space and consider the homotopy pullback diagram
$$\require{AMScd} \begin{CD} \Omega X @>>> *\\ @VVV @VVV \\ * @>>> X \end{CD}$$
Taking the induced map on cofibers, we get a diagram
$$\require{AMScd} \begin{CD} \Omega X @>>> * @>>> \Sigma \Omega X\\ @VVV @VVV @VVfV\\ * @>>> X & @>>> X \end{CD}$$
The map $f$ is the counit of the $\Sigma$-$\Omega$ adjunction. Is there a formal way to prove this (i.e. without factoring the map $\Omega X \to *$ explicitly as a cofibration followed by a weak equivalence)?