An equivalence of categories of presheaves. Let $C$ and $D$ be two small categories. Consider the corresponding categories of presheaves $PSh(C)$ and $PSh(D)$.
Suppose we have an equivalence of categories $F: PSh(C) \to PSh(D)$. Asking for an induced equivalence of categories between $C$ and $D$ seems too much.
However, is it possible to derive another relation on the categories $C$ and $D$, via $F$?
 A: Yes. $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ and $[\mathcal{D}^\mathrm{op}, \mathbf{Set}]$ are equivalent if and only if the Cauchy-completions of $\mathcal{C}$ and $\mathcal{D}$ are equivalent; more precisely, given an equivalence $F : [\mathcal{C}^\mathrm{op}, \mathbf{Set}] \to [\mathcal{D}^\mathrm{op}, \mathbf{Set}]$, there exist functors
$$\mathcal{C} \rightarrow \mathcal{E} \leftarrow \mathcal{D}$$
such that $\mathcal{E}$ is Cauchy-complete (= has splittings for all idempotents), the induced functors
$$[\mathcal{C}^\mathrm{op}, \mathbf{Set}] \leftarrow [\mathcal{E}^\mathrm{op}, \mathbf{Set}] \rightarrow [\mathcal{D}^\mathrm{op}, \mathbf{Set}]$$
are equivalences, and $[\mathcal{E}^\mathrm{op}, \mathbf{Set}] \to [\mathcal{D}^\mathrm{op}, \mathbf{Set}]$ is isomorphic to the composite $[\mathcal{E}^\mathrm{op}, \mathbf{Set}] \to [\mathcal{C}^\mathrm{op}, \mathbf{Set}] \stackrel{F}{\to} [\mathcal{D}^\mathrm{op}, \mathbf{Set}]$. In particular, if $\mathcal{D}$ is already Cauchy-complete, we can take $\mathcal{D} = \mathcal{E}$ and $\mathcal{D} \to \mathcal{E}$ to be the identity.
See also here and here.
