Some adjoints in homotopy categories

Let 𝑅 be a ring. In a previous paper [11] we found a new description for the category 𝐊⁡(𝑅-Proj); it is equivalent to the Verdier quotient 𝐊⁡(𝑅-Flat)/𝒮, for some suitable 𝒮 ⊂𝐊⁡(𝑅-Flat). In this article we show that the quotient map from 𝐊⁡(𝑅-Flat) to 𝐊⁡(𝑅-Flat)/𝒮 always has a right adjoint. This gives a new, fully faithful embedding of 𝐊⁡(𝑅-Proj) into 𝐊⁡(𝑅-Flat). Its virtue is that it generalizes to nonaffine schemes.