I'm reading this entry on nLab. But I'm confusing with the notion of covering-flatness. More precisely, I meet some trouble when I try to show that the $Sets$-valued flatness is a special case of covering-flatness.

The following is a possible proof.

Let $C$ be a small category and $F\colon C\to Sets$ a covering-flat functor. To show it is $Sets$-valued flat, consider an arbitrary finite diagram $D$ in the category $El(F)$ of elements of $F$. Then this diagram itself provides a cone $\alpha\colon\Delta_\ast\to FD'$ over the image of some diagram $D'$ in $C$ with same shape of $D$ under $F$. In this way, to give a cone over $D$ is the same to give a cone $\beta\colon\Delta_U\to D'$ over $D'$ such that $\alpha$ factors through the image of $\beta$ under $F$. Now, consider the limit cone $\lambda\colon\Delta_L\to FD'$ of $FD'$, then $\alpha$ factors through it, which gives rise to an element $a$ of $L:=\lim FD'$. Then by the describe of the sieve corresponding to the limit cone $\lambda$, there exists a cone $\beta\colon\Delta_U\to D'$ over $D'$ such that $F\circ\beta$ factors through $L$ and that the $a$ lies in the image of $U\to L$. Now, any preimage $u$ of $a$ in $U$ gives a cone $\Delta_{(u\colon\ast\to F(U))}\to D$ over $D$.

But the above argument seems failed if for instance the limit $L=\varnothing$.

What's more, consider the case that $F$ maps any object in $C$ to the empty set. Obviously it is not $Sets$-valued flat. However, consider that for any finite diagram $D\colon I\to C$, the only cone over $FD$ is induced by $\mathrm{id}_{\varnothing}$. The the sieve described in the statement corresponding to this cone is $\{\mathrm{id}_{\varnothing}\}$, which is a covering sieve in $Sets$. So, this $F$ is covering-flat but not $Sets$-valued flat. What's wrong?