Contractible cosimplicial evaluation computes diagram derived colimits

The double complex. Choose the supplied representable-sum projective resolution $G_{\bullet }\to F$ of [F1], with $G_p=0$ for $p<0$. Form the first-quadrant double complex $A_{p,q}=G_p(U_q)$ for $p,q\ge 0$, with horizontal differential induced by the resolution $G_{\bullet }$ and vertical differential induced by the cosimplicial operators of $U$, one of the two signed so that the total differential squares to zero; let $T={\mathrm{Tot}}^{\oplus }A$ be the direct-sum total complex of [F3]. Each $A_{p,q}$ is an $R$-module and each bidegree with $p+q=n$ contributes to a finite direct sum in total degree $n$.

Exact rows. For fixed $q$ the evaluation functor at $U_q$ is exact by [F1], so the row $G_{\bullet }(U_q)\to F(U_q)$ is an exact augmented complex with augmentation $F(U_q)$ in degree zero. Hence the rows of $A$ are exact except for the augmentation to the degree-zero row $q\mapsto F(U_q)$.

Exact columns. For fixed $p$ the term $G_p$ is a direct sum of representables $R_V$, and $R_V(U_{\bullet })=R[{\mathrm{Hom}}_C(U_{\bullet },V)]$ is the free $R$-module on the simplicial set ${\mathrm{Hom}}_C(U_{\bullet },V)$. By hypothesis that simplicial set is contractible, so by the prism argument of [F2] its free chain complex is chain homotopy equivalent to $R[\mathrm{\Delta }[0]]$. The latter has $R$ in every nonnegative degree, differential zero in odd degrees and identity in positive even degrees; its augmentation to $R$ induces an isomorphism on $H_0$, and its positive homology vanishes. The augmentation of $R_V(U_{\bullet })$ therefore induces $H_0\cong \mathrm{colim}{R}_V=R$, and the augmented column is acyclic. Summing over the direct summands, the column $G_p(U_{\bullet })\to \mathrm{colim}{G}_p$ is acyclic in positive degrees, with $H_0=\mathrm{colim}{G}_p$.

The roof and its quasi-isomorphisms. The augmentations of steps 2.1 and 2.2 give maps of complexes $F(U_{\bullet })\leftarrow T\to \mathrm{colim}{G}_{\bullet }$, hence a roof in $D(R)$. Both maps are quasi-isomorphisms by finite-diagonal elimination: the cone of the map to $F(U_{\bullet })$ is, up to shift and sign, the total of the horizontally augmented rows. In a total cycle, the component of largest $q$ is a horizontal cycle; exactness of the row in step 2.1 supplies a horizontal lift. Subtract its total boundary, eliminating that component and introducing terms only at $q-1$, and repeat down to $q=0$. This proves the cone acyclic. For the second map use the vertically augmented columns and eliminate the component of largest $p$ by step 2.2, introducing terms only at $p-1$. The augmented indices have lower bound $-1$, and each degree has finitely many bidegrees, so both eliminations terminate.

Identification with the derived colimit, canonically. By [F1] the complex $\mathrm{colim}{G}_{\bullet }$ computes $L{\mathrm{colim}}_{C^{\mathrm{o}\mathrm{p}}}F$. Composing the two quasi-isomorphisms of step 3.1 identifies $F(U_{\bullet })$ with it in $D(R)$. Two choices of projective resolution are compared by comparison chain maps lifting the identity, and the resulting roofs agree by [F4] and its uniqueness statement, so the identification is canonical: it does not depend on the supplied replacement, and it is natural in $F$ because comparison lifts are natural and unique up to homotopy. The contraction choices of step 2.2 were used only to obtain the quasi-isomorphisms and do not enter the resulting canonical isomorphism.