A category $I$ is filtered iff any colimits of $I$ in the category of sets commutes with any finite limits A category is said to be filtered if:

*

*it is not empty

*for any objects $i, j$, there exists an object $k$ and
maps $i \to k, j \to k $.

*for any objects $i, j$ and for any maps $u, v : i \to j $, there exists $j \to k$ such that the two maps $i \to k$ coincides.

The “only if” part is fundamental.
I want to show the “if” part.
The condition 2 is easy: it follows from the commutativity of filtered colimits with direct products.
But I can’t show 1 and 3.
I think that 3 follws from the commutativity with equalizers.
And 1. seems to be wrong:
The empty category satisfies the condition of the title, but does not satisfies the condition 1.
 A: Here's a general result, which covers the case of filtered colimits and finite limits.
Proposition. Let $\mathcal{I}$ and $\mathcal{J}$ be small categories. If colimits of shape $\mathcal{I}$ commute with limits of shape $\mathcal{J}$ in $\textbf{Set}$, then  every diagram $\mathcal{J}^\textrm{op} \to \mathcal{I}$ admits a cocone in $\mathcal{I}$.
Proof. Let $F : \mathcal{J}^\textrm{op} \to \mathcal{I}$ be a diagram and consider the diagram $\mathcal{I} (F -, -) : \mathcal{J} \times \mathcal{I} \to \textbf{Set}$.
We have a comparison map:
$$\textstyle \varinjlim_\mathcal{I} \varprojlim_\mathcal{J} \mathcal{I} (F -, -) \longrightarrow \varprojlim_\mathcal{J} \varinjlim_\mathcal{I} \mathcal{I} (F -, -)$$
For each object $i$ in $\mathcal{J}$, $\varinjlim_\mathcal{I} \mathcal{I} (i, -) \cong 1$.
(This is basically a consequence of the Yoneda lemma.)
Hence, the RHS is $\varprojlim_\mathcal{J} 1$, which is just $1$.
On the other hand, the LHS is a quotient of the disjoint union of $\varprojlim_\mathcal{J} \mathcal{I} (F -, j)$ as $j$ varies, so if the comparison map is a bijection  then there exists some object $j$ in $\mathcal{J}$ for which $\varprojlim_\mathcal{J} \mathcal{I} (F -, j)$ is not empty.
But $\varprojlim_\mathcal{J} \mathcal{I} (F -, j)$ is the set of cocones from $F$ to $j$, so we are done. ■
The above argument actually only uses the hypothesis that the comparison map is a surjection. If we use the hypothesis that the comparison map is a bijection, we get a stronger conclusion:
Proposition. Let $\mathcal{I}$ and $\mathcal{J}$ be small categories. If colimits of shape $\mathcal{I}$ commute with limits of shape $\mathcal{J}$ in $\textbf{Set}$, then for every diagram $F : \mathcal{J}^\textrm{op} \to \mathcal{I}$, the category ${}^{F /} \mathcal{I}$ of cocones under $F$ is (inhabited and) connected.
Finally, a word of warning: these are necessary conditions, but not sufficient.
