Higher Yoneda Ext agrees with derived Ext

Fix $P_{\bullet }\to M$ and an extension $E$ with inclusion $i:N\to E_{n-1}$. By [F2], successively lift the augmentation to $u_k:P_k\to E_k$ for $0\le k<n$, respecting differentials. Exactness makes $u_{n-1}{d}_n$ factor uniquely as $ic$ for $c:P_n\to N$. Since $i$ is monic and $d_n{d}_{n+1}=0$, $c{d}_{n+1}=0$, so $c$ is a cocycle.

A cocycle $c:P_n\to N$ factors uniquely as $c=f{\bar{d}}_n$, where ${\bar{d}}_n:P_n\twoheadrightarrow {\mathrm{\Omega }}^{n}M=\mathrm{im}{d}_n$; its kernel is $\mathrm{im}{d}_{n+1}$. Push out $0\to {\mathrm{\Omega }}^{n}M\stackrel{j}{\to }{P}_{n-1}\to \cdots \to P_0\to M\to 0$ along $f$. The new first middle object is $E(f)=\mathrm{coker}((f,-j):{\mathrm{\Omega }}^{n}M\to N\oplus P_{n-1})$. Its inclusion of $N$ is monic by [F2], and its cokernel is the unchanged cokernel of $j$, so this is an exact $n$-extension.

Two lift systems $u,v$ in step 1.1 have the same terminal cocycle class. Indeed, projectivity and exactness construct homotopy components $h_k:P_k\to E_{k+1}$ through $k=n-2$ satisfying $u_k-v_k=d_E{h}_k+h_{k-1}{d}_P$, with $h_{-1}=0$. At $k=n-1$ the remaining difference factors uniquely through $i$ as $it$, with $t:P_{n-1}\to N$. Applying $d_n$ then gives $c-c^{\mathrm{\prime }}=t{d}_n$. For $n=1$ this last factorization is the entire argument. A chain map of extensions fixing endpoints carries one lift system to another with the same cocycle, so the class is constant on the generated equivalence relation.

If $c^{\mathrm{\prime }}=c+t{d}_n$, then $f^{\mathrm{\prime }}=f+tj$. The automorphism $(a,p)\mapsto (a-tp,p)$ of $N\oplus P_{n-1}$ carries the relation $(f,-j)$ to $(f^{\mathrm{\prime }},-j)$, so it induces an isomorphism $E(f)\to E(f^{\mathrm{\prime }})$, identity on both endpoints and on the unchanged tail. This notation denotes a biproduct matrix, and requires no element description of the abelian category. Thus step 1.2 descends from cocycles to cohomology classes.

For an extension and lifts from step 1.1, the map $(i,u_{n-1}):N\oplus P_{n-1}\to E_{n-1}$ kills $(f,-j)$, since $if=u_{n-1}j$ after canceling the epimorphism ${\bar{d}}_n$. It induces a chain map from the pushout extension back to the original, with identity endpoints. Conversely the canonical map $P_{n-1}\to E(f)$, together with the identity maps on the remaining $P_k$, is a lift system for the pushout extension and has terminal cocycle $f{\bar{d}}_n=c$. The two constructions are therefore inverse on classes.

A map $N\to N^{\mathrm{\prime }}$ sends $c$ to its postcomposition and sends $E(f)$ to its endpoint pushout. A map $M^{\mathrm{\prime }}\to M$ has a comparison lift $P(M^{\mathrm{\prime }})\to P(M)$ by [F3]; composing the lift system in step 1.1 with it computes the endpoint pullback, with cocycle obtained by precomposition. Homotopic comparison maps induce cochain-homotopic Hom maps by $s^k(\varphi )=\varphi h_{k-1}$, so these cohomology maps are independent of the comparison. In particular comparisons over identity give resolution independence. This proves both-variable naturality. Direct sums of lift systems give direct sums of cocycles; diagonal pullback followed by codiagonal pushout sends $(c,c^{\mathrm{\prime }})$ to $c+c^{\mathrm{\prime }}$. Hence the bijection respects Baer addition and transports the abelian group laws to extension classes.

By [F4], the target just proved is exactly ${\mathrm{Ext}}_P^n(M,N)$, without using balanced Ext. Apply the same argument in ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$: injective coresolutions become projective resolutions, extensions reverse endpoints, and pushouts become pullbacks. Translating back yields the asserted natural identification with ${\mathrm{Ext}}_I^n(M,N)$ and its addition.