Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $𝐊\left(R\text{-}\mathrm{Proj}\right)$ of projective $R$-modules. We produced a set of generators for this category, proved that the category is ${\aleph }_1$-compactly generated for any ring $R$, and showed that it need not always be compactly generated, but is for sufficiently nice $R$. We furthermore analyzed the inclusion $j_{!}^{}:𝐊\left(R\text{-}\mathrm{Proj}\right)\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ and the orthogonal subcategory $𝒮={𝐊\left(R\text{-}\mathrm{Proj}\right)}^{\perp }$. And we even showed that the inclusion $𝒮\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ has a right adjoint; this forces some natural map to be an equivalence...