Can someone explain why do we consider solitons as a composite object? I know that there are dual theories which the role of fundamental and solitonic objects can be mapped to each other, but I can't see for example how solitonic D1-brane is a composite object and how it is constructed from fundamental strings (F1-strings) or in the other theories (for example Montonon-Olive duality for Yang-Mills theory). I consider proton as a composite object, but in this case, it is not clear for me.
This is an all but impossibly broad question, where only a schematic cartoon might help.
Solitons are usually topologically stable lumps of a given size underpinned by a classical solution in the relevant field theory. They cannot be included in the strict Wightman axiom framework of QFT.
Since they have a characteristic size, they are thought of as some type of collective feature of QFT, and hence "composite". Solitons such as magnetic monopoles or skyrmions do exhibit particle-like features (the latter model baryons) but always in the context of a characteristic nontrivial size, and so could be loosely pictured as some type of composite object. Their quantization is normally carried out semi classically around the classical solution. Their ultrahigh energy behavior, an essential handle of point like particles, is problematic to even define.
The most celebrated duality (S) map is that between the 2d scalar sine-gordon model and the fermionic 2d massive Thirring model the prototype of bosonization. Normally, the intricate topology of the soliton solutions dictates the altered statistics required in this fermion-boson transition.