Why must cholesterol, although lipid-soluble, cross most membranes by means of receptor-mediated endocytosis or transporters? propose a mechanism for co-transport of cholesterol through a plasma membrane?

Because cholesterol is only minimally soluble in water (dissolves into the water at small concentrations) it needs to be transported as part of the complex with proteins-lipoproteins. Thus, lipoproteins bound by specific receptors on cell membranes. Lipoproteins, particles that transport cholesterol through the blood, contain lipid-soluble part (located inside the particle) and amphipathic or water-soluble part (one the surface of the particle). There are five types of lipoproteins: chylomicrons, very-low-density lipoprotein (VLDL), intermediate-density lipoprotein (IDL), low-density lipoprotein (LDL), and high-density lipoprotein (HDL). Receptors for the cholesterol (lipoprotein) are located on the surface of a plasma membrane and after the binding of lipoprotein, coated endocytic vesicles are formed from the surface. Those vesicles carry lipoprotein into the cell together with the receptor (internalization). After internalization, the receptors dissociate and recycle to the cell surface.