Projective relations transport the actual universal Chern operations

In the quotient-projective relation, the comparison takes $\xi =c_1(\mathcal{O}(1))$ to the target $\xi$, by [F1], and preserves its products and pullbacks. It takes the relation to the same monic relation. Apply coefficient uniqueness, including its stack-level projective-bundle version supplied in [F2], to identify the actual coefficients on each $BG{L}_n$. These are equal classes of maps of the entire classifying stack, not merely equal values on isomorphism classes of bundles over affine schemes. For $i\ge 1$ both universal maps are pointed, and the split basepoint evaluation into the abelian derived target makes forgetting pointing injective on these mapping classes, as in [F3]. Thus their finite-rank pointed mapping classes agree. Passing between line and quotient projectivizations uses the dual bundle, with the same Picard operation.

Fix $i\ge 1$ and take the two existing stable pointed universal maps on $BG{L}_{\mathrm{\infty }}$ supplied by [F2], placing their targets in one coefficient theory using [F1]. Step 1.1 identifies every finite-rank restriction. Apply [F3]: the odd reduced cohomology restriction system is surjective, its obstruction vanishes, and the two stable maps therefore admit a homotopy in the pointed sheaf category. This supplies the missing stable homotopy, with all site-restriction coherences, before completion. No claim that arbitrarily chosen finite-rank homotopies assemble has been used.

Apply the functorial completion construction of [F2] to the stable maps and the homotopy in step 2.1. Complete, project away from trivial rank, and use the abelian target's completion equivalence. Coefficient naturality identifies the resulting completed maps. Transport through the same natural local perfect-$K$-space comparison and its canonical local-replacement unit. This gives a homotopy of the pointed perfect-$K$-space operations as natural transformations on the smooth site.

For a closed subset $Z\subseteq V$, evaluate the single natural homotopy of step 3.1 on the restriction arrow $V\to V\setminus Z$. It is a homotopy of diagram maps, not two independently selected homotopies at its ends. Applying the homotopy-fiber functor gives the comparison of the supported operations, and the fiber projections give its forget-support square. Retain [F2]'s inverse-limit extension if an open complement requires it; no extra category-localization identification is inferred. The separate index-zero identity follows from preservation of the unit: $c_0=1$ is the unpointed series constant and the positive pointed construction is not applied to it. This proves every asserted map comparison with its original smooth scope.