STATEMENT: Proof. Let $(I,≼)$ be a directed set, and let $\left\{Mi\right\}i∈I$ be a directed system of R-modules, with $\left\{f_{ji}\right\}_{i∈I},i≼j$ a corresponding directed family of R-homomorphisms. Define $M$ to be the direct sum of the $M_i$, $$M = \bigoplus M_i\;\;\;\;\;\;i∈I$$ Now let $N$ be a submodule of M which is generated by elements $x_{ij}, i ≼ j$, which has component $x ∈ M_i$ in position $i$, and component $−f_{ji}(x) ∈ M_j$ in position $j$, and $0$ in all other positions, where we range over all $x ∈ Mi, i,j ∈ I, i ≼ j$. That is, if $ι_i : M_i → M$ is the natural injection map, then $N$ is generated by all elements of the form $x_{ij} =ι_i(x)−ι_j(f_{ji}(x)),\text{for}\;x∈Mi,i,j∈I,i≼j$. Now let $M/N$ be the quotient module, where $p : M → M/N$ is the projection map, and for each $i∈I$,let $f_i :M_i →M/N$ be defined by $f_i =p◦ι_i$. The claim is that $(M/N,(f_i)i∈I)$ is a direct limit. Given any object $(B,(h_i)i∈I)$, we must show that there is a unique R-module homomorphism $φ : M/N → B$ such that $φ ◦ f_i = h_i$ for every $i ∈ I$. Given $x ∈ M_i$, the only choice is that we must have $φ(ι_i(x) + N) = h_i(x)$. This is well defined, since $h_i(x) = h_j ◦ f_{ji}(x)$ by definition, and this extends by linearity to a homomorphism on all of $M/N$. Thus $M/N$ is the desired direct limit.
QUESTION: How does the author conclude the well-definededness of $\varphi$. I can't seem to make sense of the reasoning. Anyone mind explaining that part of the proof in a different manner, or on a more step-by-step basis. Thanks.
Let $y$ be such that $\iota_i(x)-\iota_j(y)\in N$. Then $\varphi(\iota_i(x)-\iota_j(y))=h_i(x)-h_j(y)=0$. For since $\iota_i(x)-\iota_j(y)\in N$, $y=f_{ji}(x)$, and it was given that $h_i=h_jf_{ji}$, so $h_i(x)=h_j f_{ji}(x)=h_j(y)$ and $h_i(x)-h_j(y)=0$. Thus $\varphi$ sends $N$ into $0$ and is well defined on the quotient $M/N$.