3 problems

Preservation conjecture. The functor $\textrm{Lan}_{p}p\circ-$ sends left adjoint bicomodules to left adjoint bicomodules and preserves the ternary factorization system…

Conjecture. A $n$-functor is an e.d.m. in the category of all $n$-categories if and only if it is surjective both on vertically composable triples of $n$-cells, and on horizontally…

Let $F\colon \mathsf{Set}\to\mathsf{Set}$ be an endofunctor, and consider the category of reachable pointed $F$-coalgebras with the trivial $({\mathsf{Mor}},{\mathsf{Iso}})$-factor…