= Connected components of discrete-opfibration pullbacks
{title2=$\pi_0(A\times_D B)\cong\pi_0(A)\times_{\pi_0(D)}\pi_0(B)$}
For <discrete opfibrations> $A\to D\leftarrow B$ with <weakly filtered categories>, the map taking a pair to its <connected components of a category> is a <bijection>. For surjectivity, send two base objects in the same component to a common target and lift both arrows. For injectivity, choose common targets upstairs for two representatives; a cocone over the resulting finite connected diagram downstairs makes their lifted composites agree. Hence the representatives have a common target in the <pullback in a category>. Over a <filtered category>, the component functor on the slice of discrete opfibrations preserves all finite <categorical limits>: it preserves pullbacks and takes the terminal object to a singleton.
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