Pullback and pushout of an extension

Let

$$\xi :0⟶N\stackrel{i}{\to }E\stackrel{p}{\to }M⟶0$$

be an extension in an abelian category.

For a morphism $f:M^{\mathrm{\prime }}\to M$, the pullback extension $f^{\ast }\xi$ is $0⟶N\stackrel{i^{\mathrm{\prime }}}{\to }E{\times }_M{M}^{\mathrm{\prime }}\stackrel{q_2}{\to }{M}^{\mathrm{\prime }}⟶0,$ where $q_1:E{\times }_M{M}^{\mathrm{\prime }}\to E$ and $q_2:E{\times }_M{M}^{\mathrm{\prime }}\to M^{\mathrm{\prime }}$ are the pullback projections of $p$ and $f$, and $i^{\mathrm{\prime }}$ is the unique map satisfying $q_1{i}^{\mathrm{\prime }}=i$ and $q_2{i}^{\mathrm{\prime }}=0$. Thus pullback is contravariant in the quotient variable.

For a morphism $g:N\to N^{\mathrm{\prime }}$, the pushout extension $g_{\ast }\xi$ is $0⟶N^{\mathrm{\prime }}\stackrel{j_{N^{\mathrm{\prime }}}}{\to }{N}^{\mathrm{\prime }}{⨿}_{N}E\stackrel{p^{\mathrm{\prime }}}{\to }M⟶0,$ where $j_E:E\to N^{\mathrm{\prime }}{⨿}_{N}E$ and $j_{N^{\mathrm{\prime }}}:N^{\mathrm{\prime }}\to N^{\mathrm{\prime }}{⨿}_{N}E$ are the pushout maps of $i$ and $g$, and $p^{\mathrm{\prime }}$ is the unique map satisfying $p^{\mathrm{\prime }}{j}_E=p$ and $p^{\mathrm{\prime }}{j}_{N^{\mathrm{\prime }}}=0$. For modules the pushout is $(N^{\mathrm{\prime }}\oplus E)\mathrm{/}⟨(g(n),-i(n)):n\in N⟩$. Thus pushout is covariant in the subobject variable. The two displayed sequences are exact because pullbacks preserve kernels of epimorphisms and pushouts preserve cokernels of monomorphisms in an abelian category.