Complete instrument agreement versus record agreement

A finite instrument is a family of completely positive maps $\{{\mathcal{I}}_w{\}}_w$ whose sum is trace preserving. Its effects are $E_w={\mathcal{I}}_w^{\ast }(I)$. Its recorded channel returns the classical outcome and the residual output.

Let $\{P_z{\}}_z$ be a projective measurement on an input space $H$, let $q$ aggregate its fine outcomes, and let $T:L(H)\to L(K)$ be a channel. Define

and two recorded residual channels

where $\{{\mathcal{I}}_w{\}}_w$ is an instrument on $K$.