Lax orthogonal factorisations in monad-quantale-enriched categories

We show that, for a quantale $V$ and a $\mathsf{Set}$-monad $\mathbb{T}$laxly extended to $V$-$\mathsf{Rel}$, the presheaf monad on the category of$(\mathbb{T},V)$-categories is simple, giving rise to a lax orthogonalfactorisation system (lofs) whose corresponding weak factorisation system hasembeddings as left part. In addition, we present presheaf submonads and studythe LOFSs they define. This provides a method of constructing weakfactorisation systems on some well-known examples of topological categoriesover $\mathsf{Set}$.Comment: 13 pages. Minor changes from previous versio