  Expanding on the comprehensive factorization of functors internal to a
category C, under fairly mild conditions on a monad T on C we establish that
this orthogonal factorization system exists even in Burroni's category Cat(T)
of (internal) T-categories and their functors. This context provides for some
expected applications and some unexpected connections. For example, it lets us
deduce that the comprehensive factorization is also available for functors of
Lambek's multicategories. In topology, it leads to the insight that the role of
discrete cofibrations is played by perfect maps, with the comprehensive
factorization of a continuous map given by its fibrewise compactification.
