Change-of-injective-resolution isomorphisms satisfy identity and cocycle laws

For any ordered pair $(S,T)$, [L1] gives comparison extensions $c_A:S^{\bullet }(A)\to T^{\bullet }(A)$ and $d_A:T^{\bullet }(A)\to S^{\bullet }(A)$ of $1_A$. Their composites extend $1_A$, so [L2] and [L3] show that the induced cohomology maps are inverse. Any other choice of $c_A$ extends the same identity and hence induces the same map. For a morphism $u:A\to B$, choose within-data comparison extensions using [L4]. The two composites from $S^{\bullet }(A)$ to $T^{\bullet }(B)$ both extend $u$, so [L2] and [L3] give the naturality square. Thus the displayed construction specifies a well-defined natural isomorphism ${\theta }_{S,T}$.

For the pair $(I,I)$, the identity cochain map on $I^{\bullet }(A)$ is a comparison extension of $1_A$. Any comparison extension used in step 1.1 to define ${\theta }_{I,I}(A)$ extends the same identity morphism, so [L2] makes it cochain-homotopic to the identity. By [L3], the induced map on cohomology is therefore the identity on $R_I^{n}F(A)$.

For the triple $(I,J,K)$, the cochain map defining ${\theta }_{J,K}(A)\circ {\theta }_{I,J}(A)$ is the composite of two comparison extensions of $1_A$, while the map defining ${\theta }_{I,K}(A)$ is another comparison extension of $1_A$. By [L2] these are cochain-homotopic, so [L3] gives ${\theta }_{J,K}(A)\circ {\theta }_{I,J}(A)={\theta }_{I,K}(A)\mathrm{.}$

Since $A$ was arbitrary, steps 2.1 and 2.2 prove the identity and cocycle laws for the natural isomorphisms constructed in step 1.1.