Abstract: 
We study a large class of McKeanVlasov SDEs with drift and diffusion coefficient depending on the density of the solution`s time marginal laws in a Nemytskiitype of way.
A McKeanVlasov SDE of this kind arises from the study of the associated nonlinear FPKE, for which is known that there exists a bounded Sobolevregular Schwartzdistributional solution $u$.
Via the superposition principle, it is already known that there exists a weak solution to the McKeanVlasov SDE with time marginal densities $u$.
We show that there exists a strong solution the McKeanVlasov SDE, which is unique among weak solutions with time marginal densities $u$.
The main tool is a restricted YamadaWatanabe theorem for SDEs, which is obtained by an observation in the proof of the classical YamadaWatanabe theorem. 
