I'm having trouble understanding why modifications are the right notions of $3$-morphisms. For natural transformation, we get the definition by requiring $1\text{-}Cat$ to be cartesian closed (nlab). I'm wondering if there's a similar characterization for modifications, or more general for $n$-transfors. Ideally then, I would like to have some property that $n$-transfors satisfy, that lets me compute in detail what data/relations $n$-transfors should have/satisfy.
For simplicity, I'm just wondering for the case of strict $n$-categories.