Fin pointed as spans

Remark 17.4.2. Recall from Lemma 5.3.8 that there is an equivalence \[ \Span _{\inj ,\all }(\Fin ) \simeq \Fin _*, \] given on objects by sending \(I\) to \(I_+\) and on morphisms by sending a span \(I \xhookleftarrow {f} K \xrightarrow {g} J\) to the pointed map \(h\colon I_+ \to J_+\) given by \(h(i) = g(k)\) whenever \(i = f(k)\) is in the image of \(f\) and \(h(i) = *\) otherwise. Under this equivalence, the inert maps in \(\Fin _*\) correspond to the backwards spans \(I \xhookleftarrow {f} K \xrightarrow {=} K\), and the active maps in \(\Fin _*\) correspond to the forward spans \(I \xleftarrow {=} I \xrightarrow {g} J\). The Segal maps \(\rho _i\) correspond to the backwards spans \[ I \hookleftarrow \{i\} \xrightarrow {=} \{i\}. \] We will write \(\Fin _* \hookrightarrow \Span (\Fin )\) for the inclusion of the subcategory \(\Span _{\inj ,\all }(\Fin )\).