Filtered colimits commute with finite limits in sets

In the Category of sets, filtered colimits commute with finite limits. Elements in a finite limiting diagram involve only finitely many representatives and finitely many equalities, all of which can be realized at one common stage of a filtered diagram.

In a filtered colimit in a category of sets, any finite zigzag of generating identifications can be moved to one common stage. The filtered category conditions supply common objects and equalize the finitely many competing arrows. Consequently finitely many representatives and finitely many equalities can be realized simultaneously at one stage. This is the elementwise reason filtered colimits commute with finite limits in sets.

Let $A_n=\{m\in \mathbb{N}:m\ge n\}$ with the inverse-system inclusions $A_{n+1}\hookrightarrow A_n$, and let $B_n=C_n=1$. Each pushout of $1\leftarrow A_n\to 1$ is a singleton because $A_n$ is nonempty. Yet $\underleftarrow{\lim }{A}_n=\varnothing$, so the pushout after taking limits is $1{\bigsqcup }_{\varnothing }1$, a two-element set. Thus cofiltered limits do not in general preserve pushouts.