Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such that


\begin{eqnarray}
E &\xrightarrow{f}& B\\
\small{g}\downarrow & &\|\\
E'&\xrightarrow{f'}& B
\end{eqnarray}


commutes and suppose that $g$ induces a weak equivalence $g\_x\colon E\_x\xrightarrow{\cong} E'\_x$ when restricted to the fibers $E\_x$ and $E\_x'$ of $E$ and $E′$ over the basepoint $x$ of $B$.



> 
> Is $g$ a weak equivalence?
> 
> 
> 


By the 5-lemma, the only problem is the injectivity of $\pi\_0g\colon\pi\_0E\to \pi\_0E'$. A related [question](https://math.stackexchange.com/questions/786121/why-are-the-total-spaces-of-two-serre-fibrations-equivalent-when-the-bases-and-t) on math.stackexchange does not provide an answer.



Equivalence of the total spaces of two Serre fibrations with equivalent fibers
Note that $\pi \_1(B)$ acts on $\pi \_0(F)$. So the ending of the long exact sequence for the fibration 
$$\cdots\rightarrow \pi \_1(B)\rightarrow \pi \_0(F)\rightarrow \pi \_0(E)$$
is not just an exact sequence of sets, but it also shows that $\pi \_0(E)$ is the quotient of the set $\pi \_0(F)$ by the group $\pi \_1(B)$. So your hypotheses imply that $\pi \_0(E)\cong \pi \_0(E^{\prime})$. Thus combined with what you already know about $\pi \_i$'s with $i>0$, you get the weak equivalence between
$E$ and $E^{\prime}$.


