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For a small category with finite limits, a set-valued functor preserving them has a category of elements with finite limits, so its opposite is a filtered category. The covariant density presentation is therefore a filtered colimit of representable functors. Conversely, representable functors preserve categorical limits, and filtered colimits commute with finite limits in sets, proving the criterion.

Use the covariant density presentation. An object of $\int F$ is $(c,x)$ with $x\in Fc$; an arrow $(c,x)\to (d,y)$ is $u:c\to d$ with $F(u)x=y$. For each such object the Yoneda lemma gives a natural transformation $\mathcal{C}(c,-)\to F$. An arrow $u$ induces $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$ by precomposition, so these transformations form a cocone indexed by $(\int F{)}^{\mathrm{op}}$.

At $a$, send a representative $(c,x,v:c\to a)$ to $F(v)x$. Every $z\in Fa$ comes from $(a,z,1_a)$. If two representatives have the same value $z$, their arrows to $(a,z)$ in $\int F$ identify both with that same representative in the opposite-indexed colimit. Thus the map is bijective and natural, provingThe index is small because $\mathcal{C}$ is small and every $Fc$ is a set. By (iii) it is a filtered category. This proves (iv).