Let $F: \mathcal C \to \mathcal D$ be a functor with $\mathcal D$ cocomplete, and let $\mathscr P \mathcal C$ be the free cocompletion of $\mathcal C$ (i.e., the category of small presheaves on $\mathcal C$), so that there is a (unique, up to isomorphism) cocontinuous extension $\hat F$ of $F$ along the Yoneda embedding $\mathcal C \hookrightarrow \mathscr P \mathcal C$.


I believe that the following is true: if $F$ is faithful, so then is $\hat F$. Does anyone have a nice explanation? I'd be even happier with a proof expressible in the language of enriched category theory.


A subsidiary question. The extension $\hat F$ is the left Kan extension of $F$ along the Yoneda embedding. Is that true that any left Kan extension of a faithful functor along a fully faithful functor is again faithful? Once again, an answer in enriched category theory would be very much appreciated!



Yoneda extension of a faithful functor is faithful
As for the first question, it's false. This is obvious in the case of posets $C, D$ (with $D$ a sup-lattice): any poset map $C \to D$ is faithful when considered as a functor, but the induced functor $Set^{C^{op}} \to D$ can't be faithful (for most presheaves $F, G$, there will be more than one natural transformation $F \to G$). Of course this answers the second question as well, since the assertion there is even stronger. 


