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Idea

There is a canonical way to turn any precategory into a weakly equivalent?category. This can be thought of as an analogue of univalence but for isomorphisms? instead of equivalences?.

Proof. Let $\hat{A}_0$ be the type of representable? objects of $\mathit{Set}^{A^{\mathrm{op}}}$, with hom-sets inherited from $\mathit{Set}^{A^{\mathrm{op}}}$. Then the inclusion $\hat{A} \to \mathit{Set}^{A^{\mathrm{op}}}$ is fully faithful and an embedding on objects?. Since $\mathit{Set}^{A^{\mathrm{op}}}$ is a category by Theorem 9.2.5 (see functor category?functor precategory), $\hat{A}$ is also a category. Let $A \to \hat{A}$ be the Yoneda embedding. This is fully faithful by corollary 9.5.6 (See Yoneda embedding), and essentially surjective by definition of $\hat{A}_0$. Thus it is a weak equivlance?. $\square$

$\hat{A}_0\equiv \sum_{(F : \mathit{Set}^{A^{\mathrm{op}}})} \sum_{(a:A)} \mathbf{y} a \cong F$