Analogously to this old question, I was asking myself if it is possible to describe left/right invertible natural transformations by their components. Obviously this property is inherited by the components of a transformation.

For the converse, I don't think that every choice of left inverses of the components yields a natural transformation, or even that such a choice always has to exist. My thoughts until now have been:

Given a natural transformation $\varepsilon: F \Rightarrow G$ between functors $F, G: C \to D$, if its components are right invertible with sections $\eta_C: GC \to FC$, naturality of $\eta$ comes down to $Gc \circ \eta_C = p_{C'} \circ Gc \circ \eta_C$ for any morphism $c: C \to C'$, where I defined the idempotents $p_C := \eta_C \circ \varepsilon_C$. At this point I don't see under which conditions there is a choice of $\eta_C$ and $\eta_{C'}$ such that the above equation is fulfilled (apart from $p_C = \text{id}_{FC}$, i.e. natural isomorphisms, of course).

I also don't see a way to apply the solution of the question mentioned above, since being a section/retraction is not a property that can be detected using (co)limits as far as I know (apart from "boring" categories, where the sections are exactly the effective monomorphisms, or even all monomorphisms).

Another approach might be that the split monics/epics are exactly the absolute monics/epics, but I didn't get very far this way...

If it turns out that not every natural transformation with left/right invertible components is left/right invertible, I'd be interested if there is an additional criterion on the components that guarantees the existence of a left/right inverse of the transformation (and is ideally equivalent to it).