Corollary 4

If each module $R_i,\ i = 1, ..., n,$ is injective (resp. projective), the mapping $\alpha \mapsto \theta \circ \alpha$ (resp. $\alpha \mapsto \alpha \circ \theta$) from $\mathrm{Ext}_A^m(P, M)$ to $\mathrm{Ext}_A^{m+n}(P, N)$ (resp. from $\mathrm{Ext}_A^m(N, P)$ to $\mathrm{Ext}_A^{m+n}(M, P)$) is bijective for every $A$-module $P$ and every integer $m > 0$.

This indeed follows from Corollary 3 and the exact sequences

(resp. $\mathrm{Ext}_A^{m+i-1}(R_i, P) \to \mathrm{Ext}_A^{m+i-1}(K_i, P) \to \mathrm{Ext}_A^{m+i}(K_{i-1}, P) \to \mathrm{Ext}_A^{m+i}(R_i, P)$), whose extreme terms are zero by assumption.