Suppose $(\mathcal{C},\mathcal{W})$ is a relative category (we can assume $\mathcal{C}$ is small for the matter). Is there any work which deal with constructing a relative category structure on $\mathbf{P}(\mathcal{C})$ - the category of $\mathbf{Set}$-valued presheaves on $\mathcal{C}$ extending the structure on $\mathcal{C}$?