In this talk we will study the lax orthogonal factorization systems (LOFSs) of Clementino and Franco, with a particular focus on finding equivalent definitions of them.

In particular, we wish to define them as a pair of classes ℰ and ℳ subject to some conditions. To achieve this, we will reduce the definition of a LOFS in terms of algebraic weak factorization systems (defined as a KZ 2-comonad L and KZ 2-monad R on the 2-category of arrows [2, 𝒞] with a 2-distributive law LR ⇒ RL) to a more property-like definition (meaning a definition with less data but more conditions).

To do this, we replace strict KZ 2-monads with the property-like definition of KZ pseudomonads in terms of Kan-extensions due to Marmolejo and Wood. In addition, pseudo-distributive laws involving KZ pseudomonads have a property-like description which will be used. Thus one can deduce the conditions the classes ℰ and ℳ must satisfy.

We will also consider some similarities and differences between LOFSs and (pseudo-)orthogonal factorization systems, and will extend their definitions to include universal fillers for squares which only commute up to a comparison 2-cell.