Chain complexes as a model category?

Question

I'm reading a paper called Model Categories and Simplicial Methods by Paul Goerss and Kristen Schemmerhorn, and they show that cofibrations have lifting property with respect to acyclic fibrations for chain complexes in Proposition 1.1, but I don't understand the proof. More specifically, how does the commutative diagram with fibered product imply we have a chain map?

Answer 1

At the point where Lemma 1.2 is used you are already given maps $g_k: B_k\to M_k$ for $k<n$ such that $$(\alpha): \partial^M_k g_k = g_{k-1} \partial^B_k,\quad (\beta): j_k g_k = a_k\quad\text{and}\quad (\gamma): f_k g_k = b_k$$ hold, where $a_k: A_k\to M_k$ and $b_k: B_k\to N_k$ are the components of the given morphisms $A\to M$ and $B\to N$, respectively, and you are looking for a map $g_n: B_n\to M_n$ which also has properties $(\alpha)-(\gamma)$. Conditions $(\alpha)$ and $(\gamma)$ can be reformulated as finding a lift $B_n\to M_n$ in the diagram $$\begin{array}{ccc} & & M_n\ \ \ \ \ \ \ \ \ \ \ \ & \\ & & \downarrow (\partial^M_k,f_k)\\ B_n & \xrightarrow{(\partial_k^B g_{k-1}, b_k)} & M_{n-1}\times N_n\end{array},$$ but since $(\partial_k^B g_{k-1}, b_k)$ factors through $Z_{n-1}M\times_{Z_{n-1} N} N_n\hookrightarrow M_{n-1}\times N_n$, a fortiori a lift of $$\begin{array}{ccc} & & M_n\quad\quad\ \ \ & \\ & & \downarrow (\partial^M_k,f_k)\\ B_n & \xrightarrow{(\partial_k^B g_{k-1}, b_k)} & Z_{n-1}M\times_{Z_{n-1}M} N_n\end{array}.$$ would do. Finally, condition $(\beta)$ is taken into account by adding $A_n$ in the upper left corner.