I have a question about the short five lemma. Since it is It is usually phrased something like:
Let the following be a commutative diagram of A-modules with exact rows
$$\require{AMScd} \begin{CD} 0 @>>> A & @>\psi>> B @>\phi>> C @>>> 0 \\ @. @VV\alpha_1V @VV\alpha_2V @VV\alpha_3V @. \\ 0 @>>> A' @>\psi'>> B' @>\phi'>> C' @>>> 0 \end{CD} $$ If $\alpha_1$ and $\alpha_3$ are isomorphisms, then so is $\alpha_2$
If I am given a diagram like the one above where $\alpha_2$ is not a homomorphism and just some map, does the conclusion still follow? In other words, if I have a diagram $$ \begin{CD} 0 @>>> A & @>\psi>> B @>\phi>> C @>>> 0 \\ @. @VV\alpha_1V @VV\alpha_2V @VV\alpha_3V @. \\ 0 @>>> A' @>\psi'>> B' @>\phi'>> C' @>>> 0 \end{CD} $$ where $\alpha_1$ and $\alpha_3$ are homomorphisms and $\alpha_2$ is any map (i.e. we do not assume it to be a homomorphism), does the lemma still hold? Or do we always assume $\alpha_1,\alpha_2$ and $\alpha_3$ to be homomorphisms from the start?