Consider a fiber sequence of spaces
$$F \overset{i}{\to} E \to B$$
The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point set topology) that the fiber of that map is homotopy equivalent to $\Omega B \ast F$ hence we obtain a new fiber sequence
$$\Omega B \ast F \to C(i) \to B$$
To demonstrate the power of this statement consider the case of the path space fibration where $B= X , E = PX \cong \ast , F = \Omega X$. The new fiber sequence we obtain would be
$$\Sigma (\Omega X \wedge \Omega X) \cong \Omega X \ast \Omega X \to \Sigma \Omega X \to X$$
This implies immediately that whenever $X$ is $n$-connective for $n\ge 1$ then the canonical map $\Sigma \Omega X \to X$ is ($2n-1$)-connective (which is sort of the Eckman Hilton dual of Freudenthal suspension theorem).
Question: Is there an Eckman-Hilton dual to this statement?
To be precise, let $X \to Y \overset{\pi}{\to} Z$ be a cofiber sequence and let $F := fib(\pi)$ be the fiber $\pi$. Again there's a canonical (upto homotopy) map $f: X \to F$ and we may take its cofiber to obtain a new cofiber sequence
$$X \to F \to C(f)$$
Is there a "closed formula" for $C(f)$ (and for the maps involved) in terms of $X,Y$ and $Z$ (and the maps) similar to the one from above?