Are there necessary/sufficient conditions for a functor $f\colon \mathcal C\to \widehat{\mathcal A}$ to induce an equivalence $\text{Lan}\_yf=F\colon \widehat{\mathcal C}\leftrightarrows \widehat{\mathcal A}$, so that $\cal A,C$ are Morita equivalent categories?



Morita equivalence via Kan extension
Expanding on Zhen's comment into an answer, the Cauchy completion of $A$ can be identified with the small-projective objects in $\widehat{A}$ (i.e. those where mapping out of them preserves all small colimits). Thus, an equivalence of presheaf categories must restrict to an equivalence of Cauchy completions. It follows that if $f:C\to \widehat{A}$ induces such an equivalence, it must land inside the small-projectives and be Cauchy-surjective onto them. In other words, it must be the case that an object of $\widehat{A}$ is small-projective iff it is a retract of an object in the image of $f$. Moreover, $f$ must be fully faithful, since $C$ embeds fully faithfully in $\widehat{C}$. Conversely, if $f$ satisfies these two conditions, then it induces an equivalence of Cauchy completions, hence of presheaf categories.


So the answer is, it is necessary and sufficient that


1. $f$ is fully faithful, and
2. an object of $\widehat{A}$ is small-projective iff it is a retract of an object in the image of $f$.


