Let $M$ and $N$ be projective $A$-modules. If we know that $f: M \to N$ is surjective and $g: N \to M$ is surjective, can we conclude that $M$ is isomorphic to $N$? More generally, if $M$ and $N$ are not necessarily projective, can we conclude that $M$ is isomorphic to $N$?
I am asking this question because it seems that in the first four lines of page 30 of the book elements of representation theory of associative algebras volumn 1 (see the picture below), it is said that g is a surjective map $P'$ to $P(M)$ and g' is a surjective map from $P(M)$ to $P'$ imply that g is a bijection. Thank you very much.