2 problems

Embedding conjecture. If $F$ has a diagonal-preserving lifting, then $F$ can be embedded in a weak pullback-preserving functor.

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…