The preconnecting arrow on cycles

Let $0\to A_{\bullet }\stackrel{i}{\to }{B}_{\bullet }\stackrel{p}{\to }{C}_{\bullet }\to 0$ be a short exact sequence of complexes in an abelian category, and fix $n\in \mathbb{Z}$. Apply Snake lemma under the weaker Stacks hypotheses to the quotient-kernel diagram of The cycle-boundary diagram associated to a short exact sequence of complexes. The kernel of its right vertical map is canonically $H_n(C)=Z_n(C)\mathrm{/}{B}_n(C),$ and the cokernel of its left vertical map is canonically $H_{n-1}(A)=Z_{n-1}(A)\mathrm{/}{B}_{n-1}(A)\mathrm{.}$ Thus the snake construction supplies a canonical morphism ${\delta }_n^{\mathrm{s}\mathrm{n}\mathrm{a}\mathrm{k}\mathrm{e}}:H_n(C)\to H_{n-1}(A)\mathrm{.}$

Let $q_n:Z_n(C)\to H_n(C)$ be the homology quotient from Homology object of a chain complex. The preconnecting arrow on cycles is the categorical composite ${\tilde{\mathrm{\partial }}}_n:={\delta }_n^{\mathrm{s}\mathrm{n}\mathrm{a}\mathrm{k}\mathrm{e}}{q}_n:Z_n(C)\to H_{n-1}(A)\mathrm{.}$

In a module category, applying this morphism to an element gives the usual lift-and-boundary recipe. The definition above uses only kernels, cokernels, and the canonical snake morphism, so it is valid in every abelian category.