Lemma 22.3.9 (Animae as Ind-completions). The restricted Yoneda functors induce equivalences \[ \An \iso \Ind (\An ^{\fin }) \qquad \text {and}\qquad \An _*\iso \Ind (\An _*^{\fin }). \]
Proof. The first equivalence follows immediately from Proposition 4.3.11, Proposition 22.3.6. For the second, Lemma 4.1.13 gives \[ \Fun ^{\lex }((\An _*^{\fin })\catop ,\An ) \iso \Fun ^{\lex }((\An _*^{\fin })\catop ,\An _*), \] and the right-hand side is equivalent to \(\An _*\) by Lemma 4.3.12. In both cases the inverse to evaluation is the restricted Yoneda functor, so the result follows from Proposition 22.3.6. □
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