  We propose a new theory of (non-split) P^n-functors. These are F: A -> B for
which the adjunction monad RF is a repeated extension of Id_A by powers of an
autoequivalence H and three conditions are satisfied: the monad condition, the
adjoints condition, and the highest degree term condition. This unifies and
extends the two earlier notions of spherical functors and split P^n-functors.
We construct the P-twist of such F and prove it to be an autoequivalence. We
then give a criterion for F to be a P^n-functor which is stronger than the
definition but much easier to check in practice. It involves only two
conditions: the strong monad condition and the weak adjoints condition. For
split P^n-functors, we prove Segal's conjecture on their relation to spherical
functors. Finally, we give four examples of non-split P^n-functors: spherical
functors, extensions by zero, cyclic covers, and family P-twists. For the
latter, we show the P-twist to be the derived monodromy of associated Mukai
flop, the so-called `flop-flop = twist' formula.
