Cofiltered limits need not commute with finite colimits in sets
= Cofiltered limits need not commute with finite colimits in sets
Let $A_n=\{m\in\mathbb N:m\geq n\}$ with the inverse-system inclusions $A_{n+1}\hookrightarrow A_n$, and let $B_n=C_n=1$. Each pushout of $1\leftarrow A_n\to1$ is a singleton because $A_n$ is nonempty. Yet $\varprojlim A_n=\varnothing$, so the pushout after taking limits is $1\sqcup_\varnothing1$, a two-element set. Thus cofiltered limits do not in general preserve pushouts.