In page 42 of their book, there is a setence "in this case, it can be easily seen that the homomorphism "$\overline{u}$:Ker(T(M)$\rightarrow$ T(P))$\rightarrow$ T(M)" is defined by inclusion". I can not check it, can someone who read that book explain that? Thanks.
I think by a "snake lemma" type diagram chase, the homomorphism above should be multiplication by -1. The question I ask is as follows：
Let $R$ be a ring. When there is an exact sequence of left $R$-modules, $0\rightarrow A\rightarrow P\xrightarrow{f} B\rightarrow 0$ with $P$ a projective $R$-module, we can form exact sequences $0\rightarrow A\xrightarrow{f_1} M\xrightarrow{g_1} P\rightarrow 0$, and $0\rightarrow A\xrightarrow{f_2} M\xrightarrow{g_2} P\rightarrow 0$, where $M$ is a submodule of $P\oplus P$ consisting of all pairs $(p_1,p_2)$ such that $f(p_1)=f(p_2)$, and $f_1(a)=(a,0),g_1(p_1,p_2)=p_2$ $f_2(a)=(0,a),g_2(p_1,p_2)=p_1$. If T is a covariant additive functor, does $T(f_1)=T(f_2)$, or $T(f_1)=-T(f_2)$?