Yoneda product is composition in the derived category

For an extension $0\to N\to E_{n-1}\to \cdots \to E_0\to M\to 0$, let $K_0=M$, $K_n=N$, and use its successive images to form short exact sequences $e_j:0\to K_{j+1}\to E_j\to K_j\to 0$ for $0\le j<n$. Let $b_j:K_j\to K_{j+1}[1]$ be the connecting arrow from [F2]. Define $\mathrm{\Theta }(E)=b_{n-1}[n-1]\cdots b_1[1]b_0$. Naturality of [F2] gives a commuting diagram of these arrows for any map of extensions fixing the endpoints, so $\mathrm{\Theta }$ is constant on the generated Yoneda equivalence relation. The same naturality gives endpoint pullback and pushout compatibility. Direct sums of the short exact pieces give direct sums of their arrows; diagonal pullback and codiagonal pushout therefore make $\mathrm{\Theta }$ additive for Baer addition.

Here is the degree-one comparison explicitly in the projective lane. For $0\to N\stackrel{i}{\to }E\to M\to 0$, lift $P_0\to M$ to $v:P_0\to E$ and write $v{d}_1=ic$ with $c:P_1\to N$. The cone of $i$ has $N$ in degree $-1$, $E$ in degree zero, and differential $i$. The map $P\to \mathrm{Cone}(i)$ with components $c,v$ in degrees $-1,0$ is a complex map over $M$. Projection to $N[1]$ gives the cocycle $c$. Thus the connecting arrow is the roof of $c$, and in particular is exactly the degree-one map used to define $\mathrm{\Theta }$, without an unproved appeal to the abstract Ext/Hom isomorphism. In the injective lane extend $N\to I^0$ to $w:E\to I^0$ and factor $dw=tq$ through $q:E\to M$. The identity $p\simeq tq$ after mapping $N[1]\to I[1]$ is witnessed on the cone by the homotopy with component $w:E\to I^0$; hence the connecting arrow is represented by $t:M\to I[1]$. This explicit sign is part of our normalization.

We verify bijectivity without assuming that an arbitrary natural Ext/Hom identification preserves products. In the projective lane let $K_j={\mathrm{\Omega }}^{j}M$ in the supplied resolution. Exactness of [F5] for $K_1\to P_0\to M\to K_1[1]$ gives ${\mathrm{Hom}}_D(M,N[1])$ as the quotient of $\mathrm{Hom}(K_1,N)$ by restrictions from $P_0$, because positive Ext from the projective $P_0$ vanishes by [F1]. The quotient map sends $h$ to $h[1]b_0$. By the pushout construction in [F4], this is $\mathrm{\Theta }$ of the pushout extension. In degree $n$, that same construction identifies Yoneda classes with $\mathrm{Hom}(K_n,N)$ modulo restrictions from $P_{n-1}$: a resolution cocycle factors through $K_n$, and a coboundary is precisely such a restriction. The degree-one quotient for $0\to K_n\to P_{n-1}\to K_{n-1}\to 0$, followed by the connecting isomorphisms ${\mathrm{Hom}}_D(K_j,N[r])\cong {\mathrm{Hom}}_D(K_{j-1},N[r+1])$ for $r>0$, identifies this quotient bijectively with ${\mathrm{Hom}}_D(M,N[n])$. Each isomorphism follows from [F5] and the vanishing of both adjacent positive Hom groups from $P_{j-1}$ by [F1]. The composite is $\mathrm{\Theta }$ of the corresponding pushout extension because its connecting arrows are the short exact pieces of that extension, as in step 1.1. This establishes bijectivity directly. The normalization is not asserted to equal [F1]'s classical-projective comparison: in degree one, step 1.2 represents $\mathrm{\Theta }$ by the positive cocycle $c$, while [F1] sends the positive classical Ext cocycle $c$ to $-c$. The sign is an invertible scalar and does not affect this bijectivity argument; product compatibility follows from the connecting-arrow construction of step 1.1 rather than from an identification of the two normalizations.

In the injective lane put $C^0=N$ and $C^{j+1}=\mathrm{coker}(C^j\to I^j)$ in the supplied coresolution. The dual construction in [F4] identifies degree-$n$ Yoneda classes with $\mathrm{Hom}(M,C^n)$ modulo maps factoring through $I^{n-1}\to C^n$. Apply [F5] to $C^{n-1}\to I^{n-1}\to C^n\to C^{n-1}[1]$ to identify this quotient with ${\mathrm{Hom}}_D(M,C^{n-1}[1])$. The subsequent connecting isomorphisms identify it with ${\mathrm{Hom}}_D(M,N[n])$, since both neighboring positive Hom groups into each injective vanish by [F1]. On the pullback extension supplied by [F4], naturality of [F2] makes the composite exactly $\mathrm{\Theta }$. Thus the same intrinsic $\mathrm{\Theta }$ is bijective in either lane, including when both are available.

The short exact pieces of a splice are the pieces of its two factors in order. The definition in step 1.1 therefore gives $\mathrm{\Theta }(\beta \ast \alpha )=\mathrm{\Theta }(\beta )[p]\mathrm{\Theta }(\alpha )$, using associativity of composition and $[p][q]=[p+q]$. This also shows that the bijection is independent of the resolution used to prove its bijectivity. Degree-zero endpoint maps act by pullback and pushout by step 1.1, and identities act as units.

Apply ${\mathrm{Hom}}_D(-,A[1])$ to the triangle $B\to Z^{\mathrm{\prime }}\to C\to B[1]$ of $e^{\mathrm{\prime }}$. Its connecting arrow is $\mathrm{\Theta }(e^{\mathrm{\prime }})$ by definition and step 1.2. By [F5] the kernel of the map from ${\mathrm{Hom}}_D(B,A[1])$ to ${\mathrm{Hom}}_D(C,A[2])$ is the image of restriction from $Z^{\mathrm{\prime }}$; the map is shifted composition with $\mathrm{\Theta }(e^{\mathrm{\prime }})$, up to an irrelevant overall rotation sign. By step 3.1 its kernel is precisely the zero-splice condition. Bijectivity and pullback naturality of $\mathrm{\Theta }$ give the asserted extension over $Z^{\mathrm{\prime }}$, in both directions. Its pullback middle object is isomorphic to $Z$ by equivalence of short extensions. The kernel of $W\to Z^{\mathrm{\prime }}\to C$ is that pullback and the composite is epic, yielding $0\to Z\to W\to C\to 0$. Conversely exactness of the stated diagram identifies $Z$ with this kernel and hence with the pullback.