Comparing $\mathrm{A}\mathrm{d}\mathrm{d}(M)$ with $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}(M)$

We present characterizations for the inclusions $\mathrm{A}\mathrm{d}\mathrm{d}(M)\subseteq \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}(M)$ and $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}(M)\subseteq \mathrm{A}\mathrm{d}\mathrm{d}(M)$ in locally finitely presented categories and in compactly generated triangulated categories. As applications, we describe the situations when the classes of the form $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}(M)$ and $\mathrm{A}\mathrm{d}\mathrm{d}(M)$ are (pre)covering, respectively (pre)enveloping.