Some statemenst about, when a functor preserves / reflect monos, regular monos, split monos are already "well known". So:
When does a functor preserve or reflect strong monos / strong epis?
I don't know whether there is any nice property implying any one of the two things. So let's ask something more specific:
- Do lex functors preserve strong monos? If not, what is a counterexample?
- What if the functor is faithful or representable, does it reflect strong monos?
- What if it is continuous or even has an adjoint?
Remark: None of the bulletpoints are part of the actual question. There are meant as guidelines or as alternate things to adress if the question in the title cannot (in any obvious way) really be answered. I'm including these because I don't know what to expect from this.
Remark 2: You can answer the / a dual question, I don't really mind or care.