The Stacks project

Proof. If $E$ is a classical generator for $\mathcal{D}_ c$, then $\mathcal{D}_ c = \langle E \rangle$. It follows formally from the assumption that $\mathcal{D}$ is compactly generated and Lemma 13.36.4 that $E$ is a generator for $\mathcal{D}$.

The converse is more interesting. Assume that $E$ is a generator for $\mathcal{D}$. Let $X$ be a compact object of $\mathcal{D}$. Apply Lemma 13.37.3 with $I = \{ 1\}$ and $E_1 = E$ to write

as in the lemma. Since $X$ is compact we find that $X \to \text{hocolim} X_ n$ factors through $X_ n$ for some $n$ (Lemma 13.33.9). Thus $X$ is a direct summand of $X_ n$. By Lemma 13.37.4 we see that $X$ is an object of $\langle E \rangle$ and the lemma is proven. $\square$