Let $\textbf{Ch}_R$ the category of chain complexes of $R-$modules ($R$ is an associative ring with unit). I want to prove that this cat. satisfies the model category axioms. In particular we want to consider a particular "class" of maps in $\textbf{Ch}_R$, called cofibration, i.e. a an arrow $f$ s.t. for each $k\geq 0$, $f_k: M_k \to N_k$ is a monomorphism with a projective $R-$module as its cokernel.
AIM: the aim is to prove that the composition of two cofibrations is again a cofibration.
MY ATTEMPT: we just work in a fixed degree, hence $f \colon M_k \to N_k$ and $g\colon N_k \to Z_k$. the composition is clearly a monomorphism, but I've some problem in proving that $\text{coker}(gf)$ is projective. My idea was to using this answer, but I cannot manage to get the s.e.s of cokernels. The first arrow seems to be $g$, but then I don't know what to put in the other two.
As you noted, my answer gives you that if you have to monomorphisms $f\colon: M \to N$ and $g\colon N \to Z$, then there is a short exact sequence $$ 0\to \operatorname{coker} f \to \operatorname{coker} gf \to \operatorname{coker} g \to 0.$$ Now if $\operatorname{coker} f$ and $\operatorname{coker} g$ both are projective, then certainly so is $\operatorname{coker} gf$.