The effacement extension commutes with connecting morphisms

In the homological case, write the given sequence as $0\to A^{\mathrm{\prime }}\to A\to A^{\mathrm{\prime }\mathrm{\prime }}\to 0$ and choose the projective effacement $0\to K\to P\stackrel{p}{\to }{A}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ used by [L1] to define $u_n(A^{\mathrm{\prime }\mathrm{\prime }})$. Projectivity of $P$ lifts $p$ through the epimorphism $A\to A^{\mathrm{\prime }\mathrm{\prime }}$; the lift restricts to a map $k:K\to A^{\mathrm{\prime }}$, producing a morphism from the effacement sequence to the given sequence. By [L2], using this ladder-compatible effacement does not change $u_n(A^{\mathrm{\prime }\mathrm{\prime }})$. Naturality of the two connecting morphisms, the defining equation from [L1], and naturality of the already constructed $u_{n-1}$ give ${\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^T{u}_n(A^{\mathrm{\prime }\mathrm{\prime }})=T_{n-1}(k){\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^T{u}_n(A^{\mathrm{\prime }\mathrm{\prime }})=T_{n-1}(k)u_{n-1}(K){\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^S=u_{n-1}(A^{\mathrm{\prime }}){\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^S$. This is the required homological connecting square.

In the cohomological case, choose the injective effacement $0\to A^{\mathrm{\prime }}\stackrel{e}{\to }I\to C\to 0$ used by [L1] to define $u^{n+1}(A^{\mathrm{\prime }})$. Injectivity of $I$ extends $e$ across the monomorphism $A^{\mathrm{\prime }}\to A$ and induces a map $A^{\mathrm{\prime }\mathrm{\prime }}\to C$, producing a morphism from the given sequence to the effacement sequence. Again [L2] permits this compatible choice. Naturality of the connectors, the defining cokernel equation from [L1], and naturality of $u^n$ then give $u^{n+1}(A^{\mathrm{\prime }}){\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^S=u^{n+1}(A^{\mathrm{\prime }}){\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^S{S}^n(A^{\mathrm{\prime }\mathrm{\prime }}\to C)={\mathrm{\partial }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^T{T}^n(A^{\mathrm{\prime }\mathrm{\prime }}\to C)u^n(A^{\mathrm{\prime }\mathrm{\prime }})={\mathrm{\partial }}_{\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}}^T{u}^n(A^{\mathrm{\prime }\mathrm{\prime }})$. This is the required cohomological connecting square.

The two cases show that the one-step extension preserves every connecting morphism, independently of the effacement choices by [L2].