Proposition. Unique factorization lifting if and only if twisted arrow functor is discrete fibration [mde-8YS0]

Let \(F : \mathbb {C} \to \mathbb {D}\) be a functor, then the following are equivalent,

  1. \(F\) is a unique lifting of factorizations functor,
  2. the induced functor \(F_t : \mathbb {C}_t \to \mathbb {D}_t\) between twisted arrow categories is a discrete fibration.

Proof.  [#249]

(1 \(\to \) 2) A morphism \((\alpha _1, \alpha _2) : q \to F_t(r)\) in the base \(\mathbb {D}_t\) amounts to a commuting square in \(\mathbb {D}\) that provides a factorization \(F_t(r) = \alpha _1;q;\alpha _2\). Using the ULF property (twice or in unbiased form), one obtains a unique lift of \((\alpha _1,\alpha _2)\).

(2 \(\to \) 1) Considering a factorization in \(\mathbb {D}\) as a morphism in \(\mathbb {D}\) into an identity, and one obtains a unique lift that is a lifting of factorizations by ULF functors have discrete fibres.