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 exactly $\mathrm{\Theta }$ 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]. Their composite is exactly the formula in step 1.1 for the pushout extension.

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.