= Solution
A <filtered category> is nonempty, every pair of objects maps to a common object, and every parallel pair becomes equal after postcomposition. Equivalently every finite <diagram in a category> has a <cocone>.
For a filtered set diagram $D:I\to\mathbf{Set}$, the <finite-stage equality in a filtered set colimit> says that representatives $x\in D(i)$ and $y\in D(j)$ define the same element exactly when their images agree at some common stage. Indeed, equality in the ordinary disjoint-union-and-quotient construction is witnessed by a finite zigzag; filteredness moves its objects to a common stage and equalizes the finitely many competing arrows.
For a finite <categorical product>, choose a common stage for all coordinate representatives to obtain surjectivity of the colimit-to-product comparison. Finitely many coordinate equalities can then be realized together at one later stage, proving injectivity. The empty product is preserved because a nonempty filtered category is connected. For an <equalizer>, an element whose two images agree in the colimit reaches a stage where they agree, so it is represented by an element of the stagewise equalizer; the same equality criterion proves injectivity. Finite products and equalizers construct every <finite limit>. Hence <filtered colimits commute with finite limits in sets>.
We prove the equivalence by the cycle <i/solution>, <ii/solution>, <iii/solution>, <iv/solution>, then back to <i/solution>.
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