The long exact Ext sequence in the first variable

Regard ${\mathrm{Hom}}_{\mathcal{A}}(-,N)$ as a left exact functor ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}\to \mathbf{A}\mathbf{b}$. A projective resolution in $\mathcal{A}$ is an injective resolution in ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$, so Right derived functors form a cohomological delta functor on the opposite category gives the displayed order and connecting maps.

Translating the short exact sequence to the opposite category gives the three Hom terms in the displayed order and the maps ${\mathrm{\partial }}^q$. The balanced Ext bifunctor identifies the right-derived groups with Ext, and delta-functor naturality gives contravariant naturality in the short exact sequence.

To check covariance in $N$, fix a projective horseshoe $0\to P^{\mathrm{\prime }}\to P\to P^{\mathrm{\prime }\mathrm{\prime }}\to 0$ for the given short exact sequence, as supplied by The horseshoe lemma for projective resolutions. Its degreewise splitting makes $0\to \mathrm{Hom}(P^{\mathrm{\prime }\mathrm{\prime }},N)\to \mathrm{Hom}(P,N)\to \mathrm{Hom}(P^{\mathrm{\prime }},N)\to 0$ a short exact sequence of cochain complexes. Postcomposition with $v:N\to N^{\mathrm{\prime }}$ gives a morphism from this sequence to the analogous one with coefficients $N^{\mathrm{\prime }}$. By Naturality of the cohomology connecting morphism, all connecting squares commute. These are the horseshoe connecting maps used by the right-derived theorem in step 1.1. The comparison isomorphisms transporting the middle horseshoe resolution to the supplied one are induced by precomposition, which commutes with postcomposition by $v$. Thus the transported connecting maps, and hence the balanced Ext sequence of step 2.1, are covariantly natural in $N$.