Meaning
A raw certified comparison records both the admitted base arrow and the full physical comparison over it.
It has the form (f;R), with f a change of context and R a source-certified comparison of the full packages. Declared two-morphisms compare such passages, and composition pastes the physical comparisons while retaining their coefficient, analytic, and validity data.
Example (illustrative)
If R compares X to Y over f and S compares Y to Z over g, their composite lies over gf. Its Hamiltonian residual is propagated by the typed composition law, rather than reset to zero.
Scope and limits
Two comparisons over the same base arrow can contain different physical information. Equality of base arrows does not identify the passages above them.
Source
GAT VIII — Indexed Admissible Distributors and Selected Cofinal Čech Transport in Semisimple Yang–Mills Theory. GAT VIII, Section 4.