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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