Span preserves limits and filtered colimits

Proof. The inclusion \(N\colon \Cat _{\infty } \hookrightarrow \sAn \) preserves limits and filtered colimits (see Lemma 24.2.2 for the latter claim). It will thus suffice to show that \(\NSpan \colon \AdTrip \to \sAn \) preserves these (co)limits. Since these are computed pointwise, this boils down to showing that the functor \[ \Hom _{\AdTrip }(\Tw ^r([n]), -)\colon \AdTrip \to \An \] preserves them. The case for limits is clear, since the functor is corepresented. For filtered colimits, we use again that filtered colimits commute with pullbacks, so that we may reduce to the cases \(n = 0\) and \(n=1\) via the Segal condition. Note that the \(\infty \)-categories \(\Tw ^r([0]) = *\) and \(\Tw ^r([1]) = \; \pushout \) are finite, so \(\Hom _{\Cat _{\infty }}(\Tw ^r([n]),-)\colon \Cat _{\infty } \to \An \) preserves filtered colimits by Corollary 24.2.3. This proves the claim for \(n = 0\). For \(n = 1\), one observes that the subanimae of morphisms of adequate triples are closed under filtered colimits. □