Example 21.48.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a perfect object of $D(\mathcal{O})$. Set $K^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, \mathcal{O})$ as in Lemma 21.48.4. Then the map
\[ K \otimes _\mathcal {O}^\mathbf {L} K^\vee \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, K) \]
is an isomorphism (by the lemma). Denote
\[ \eta : \mathcal{O} \longrightarrow K \otimes _\mathcal {O}^\mathbf {L} K^\vee \]
the map sending $1$ to the section corresponding to $\text{id}_ K$ under the isomorphism above. Denote
\[ \epsilon : K^\vee \otimes _\mathcal {O}^\mathbf {L} K \longrightarrow \mathcal{O} \]
the evaluation map (to construct it you can use Lemma 21.35.6 for example). Then $K^\vee , \eta , \epsilon $ is a left dual for $K$ as in Categories, Definition 4.43.5. We omit the verification that $(1 \otimes \epsilon ) \circ (\eta \otimes 1) = \text{id}_ K$ and $(\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{K^\vee }$.
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