Arrow category $\cal K^\to$

Question

I would like to understand here on the page $6$ in the definition $3.3$ how works the functor $F:\cal K^\to \to K$. They say

 giving the (cofibration, trivial fibration) factorization 

By this I understand that we take one arrow and factor it into two given by this factorization property. But our defined $F$ like that leads in the opposite direction: we take 2 arrows and assign it 1.

Answer 1

See the beginning of section 2 in the paper. The functor $F$ is meant to send a map to the middle object that appears in its factorisation.