The horseshoe lemma for projective resolutions

Start at degree zero with the original short exact sequence $0\to A^{\mathrm{\prime }}\to A\to A^{\mathrm{\prime }\mathrm{\prime }}\to 0.$ Every current short exact kernel sequence extends one more degree by [L1], and the next choice depends on the previously constructed degree. Therefore [L4] produces the whole augmented complex for $A$, whose degree-$n$ term is $P_n^{\mathrm{\prime }}\oplus P_n^{\mathrm{\prime }\mathrm{\prime }}$ and whose kernel sequences remain short exact in every degree.

Each term $P_n^{\mathrm{\prime }}\oplus P_n^{\mathrm{\prime }\mathrm{\prime }}$ is projective by [L3], and the short exact kernel sequences from step 1.1 are exactly the data needed for exactness of the middle augmented complex. Therefore [L2] identifies the resulting complex as a projective resolution of $A$.