Suppose $\mathscr{A}$ is a dg category and $ \mathcal{D} (\mathscr{A})$ its associated derived category. For an object in $ a \in \mathscr{A}$ there is associated (right )dg $ \mathscr{A}$ module, $ \hat{a} := \mathscr{A} (?, a)$. One can define the subcategory $ \text{Perf}(\mathscr{A})$ of $ \mathcal{D} ( \mathscr{A})$ as the smallest thick subcategory ( i.e. strict, triangulated and closed under taking direct summands) containing the set of objects $\{ \hat{a}| a \in \mathscr{A} \}$.

Suppose we take our dg category is $ \mathscr{A}:= k$ i.e. the category with one object whose endomorphism space is some commutative ring $k$. Then $\mathcal{D}( k)$ is just the derived category category of chain complexes over $k$.

Question: How does one see that an object in Perf($k$), as defined above, is a perfect chain complex in the usual sense of homological algebra, i.e. quasi-isomorphic to a bounded complex of finitely generated projectives without using the fact that these are just the compact objects? ( or is this even possible?)