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 orthogonal factorisation system (lofs) whose corresponding weak factorisation system has embeddings as left part. In addition, we present presheaf submonads and study the LOFSs they define. This provides a method of constructing weak factorisation systems on some well-known examples of topological categories over $\mathsf{Set}$.Comment: 13 pages. Minor changes from previous version

@article{ClementinoF17,
  title = {Lax orthogonal factorisations in monad-quantale-enriched categories},
  author = {Maria Manuel Clementino and Ignacio Lopez Franco},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 13, Issue 3},
  year = {2017},
  doi = {10.23638/lmcs-13(3:32)2017}
}