Let $R, S$ be rings with unity and $F : {\mathcal{M}}_R \rightarrow {\mathcal{M}}_S$ be an additive functor. Let $$ 0 \rightarrow A \xrightarrow{\alpha} B \xrightarrow{\beta} C \rightarrow 0 $$ be a a short exact sequence of right $R$-modules. The question is to construct a long exact sequence of right derived functors (which by definition, are the cohomologies after taking the functor to the injective resolutions) $$ \dotsc L^n FC \xrightarrow{\delta^n} L^{n+1} FA \xrightarrow{\alpha^{n+1 \ast}} L^{n+1} FB \xrightarrow{\beta^{n+1 \ast}} L^{n+1} FC \xrightarrow{\delta^{n+1}} L^{n+2} FA \rightarrow \dotsc $$
I am trying to mimic the opposite of "pull back" construction made while dealing with the "projective" version of the same: i.e., a "pushout" may be the right choice
$\require{AMScd}$ \begin{CD} @. 0 @. 0 @. {} @. 0 @. 0 \\ @. @VVV @VVV @. @VVV @VVV \\ 0 @>{}>> A @>{d^{-1}}>> P_0 @>{d^0}>> \dotsc @>{d^{n-3}}>> P_{n-2} @>{d^{n-2}}>> P_{n-1} @>{d^{n-1}}>> P_n @>{d^n}>> \dotsc \\ {}@V \alpha VV @V f^0 VV @. @V f^{n-2} VV @V f^{n-1} VV \\ 0 @>{}>> B @>{{d^{\prime}}^{-1}}>> Q_0 @>{{d^{\prime}}^0}>> \dotsc @>{d^{n-3}}>> Q_{n-2} @>{{d^{\prime}}^{n-2}}>> Q_{n-1}\\ {} @V \beta VV @V g^0 VV @. @V g^{n-2} VV @V g^{n-1} VV \\ 0 @>{}>> C @>{{d^{\prime\prime}}^{-1}}>> R_0 @>{{d^{\prime\prime}}^0}>> \dotsc @>{{d^{\prime\prime}}^{n-3}}>> R_{n-2}@>{{d^{\prime\prime}}^{n-2}}>> R_{n-1} \\ @. @VVV @VVV @. @VVV @VVV \\ @. 0 @. 0 @. {} @. 0 @. 0 \\ \end{CD}
This is easy to build for the $0$-th column. Assuming this is done until $(n-1)$-th stage (all rows and columns exact so far), a necessary requirement is to prove:
${\mathrm{\bf Q.}}$ Show that $\big( {\mathrm{Im}}d^{n-2} \big) f^{n-1} = {\mathrm{Im}}{d^{\prime}}^{n-2}$
This is used to construct the pushout of $P_n$ and ${\mathrm{Im}}{d^{\prime}}^{n-2}$ which embeds in a suitable injective $Q_n$.
I don't see a way to chase the diagram backwards to prove this.