Proposition: Scalar reversal

For real $c$,

If scalar descent holds on the left, the prescription

defines the corresponding right scalar map on the conjugate reverse stage, with the required calibration.

Proof. Assumption ass:calibration gives $({u_{\mu }}^{\rho }+cr{)}^{†}={u_{\mu }}^{\rho }+cr$. Pairing this identity with product tests proves the first claim. Conjugation sends ${\mathcal{L}}_2$ onto ${\mathcal{R}}_2$ by (eq:Qadjoint). Thus the displayed prescription is well defined whenever the left map is, and its value is the reversed calibration.