  A pair of quantum channels are said to be incompatible if they cannot be
realized as marginals of a single channel. This paper addresses the general
structure of the incompatibility of completely positive channels with a fixed
quantum input space and with general outcome operator algebras. We define a
compatibility relation for such channels by identifying the composite outcome
space as the maximal (projective) $C^\ast$-tensor product of outcome algebras.
We show theorems that characterize this compatibility relation in terms of the
concatenation and conjugation of channels, generalizing the recent result for
channels with quantum outcome spaces. These results are applied to the positive
operator valued measures (POVMs) by identifying each of them with the
corresponding quantum-classical (QC) channel. We also give a characterization
of the maximality of a POVM with respect to the post-processing preorder in
terms of the conjugate channel of the QC channel. We consider another
definition of compatibility of normal channels by identifying the composite
outcome space with the normal tensor product of the outcome von Neumann
algebras. We prove that for a given normal channel the class of normally
compatible channels is upper bounded by a special class of channels called
tensor conjugate channels. We show the inequivalence of the $C^\ast$- and
normal compatibility relations for QC channels, which originates from the
possibility and impossibility of copying operations for commutative von Neumann
algebras in $C^\ast$- and normal compatibility relations, respectively.
