The connecting map for left derived functors

Assume the Axiom of Dependent Choice.

Let $P$ be supplied projective resolution data on a class $\mathcal{D}$ in an abelian category $\mathcal{A}$, and let $F:\mathcal{A}\to \mathcal{B}$ be an additive right exact functor. Fix a short exact sequence $0\to A^{\mathrm{\prime }}\to A\to A^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of objects of $\mathcal{D}$.

Choose a horseshoe projective resolution $H_{\bullet }$ of $A$ whose end terms are the supplied resolutions $P_{\bullet }(A^{\mathrm{\prime }})$ and $P_{\bullet }(A^{\mathrm{\prime }\mathrm{\prime }})$. By The horseshoe construction stays short exact after applying a right exact functor and The connecting morphism in homology, this yields connecting morphisms ${\mathrm{\partial }}_n^H:H_n(F(P_{\bullet }(A^{\mathrm{\prime }\mathrm{\prime }}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}))\to H_{n-1}(F(P_{\bullet }(A^{\mathrm{\prime }}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}})),n>0.$

Replace the supplied datum $P$ only at the object $A$ by the chosen horseshoe resolution $H_{\bullet }$. The resulting datum computes naturally isomorphic left derived functors by Two supplied projective resolution data define naturally isomorphic left derived functors, so each ${\mathrm{\partial }}_n^H$ may be read as a map ${\mathrm{\partial }}_n^H:L_n^{P}F(A^{\mathrm{\prime }\mathrm{\prime }})\to L_{n-1}^{P}F(A^{\mathrm{\prime }})\mathrm{.}$

This map is the connecting map for the left derived functors attached to the chosen horseshoe resolution. The next item proves that it is independent of the horseshoe choice and of the comparison isomorphisms used to read it in the fixed datum $P$.