I have inclusion of topological spaces $f:A\to B$. Then there is a fibration $E_f \xrightarrow{} B$, where $E_f$ stands for mapping path space. As I understand, $A$ is homotopy equivalent to $E_f$. Then I look at long exact sequence of our fibration. I am given that $sk_3 A=sk_3 B$, so I conclude that $\pi_2(E_f)=\pi_2(A)=\pi_2(B).$ Is it true that map $\pi_2(E_f)\to \pi_2(B)$ in the long exact sequence is an isomorphism? It would be really cool for me, if it were, however I don't manage to prove that (and I'm not sure whether it's true).
Thank you.