Relative morphisms

Notation 14.1.5. Given a functor \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) and a morphism \(\alpha \colon I \to J\) in \(\Span (\Fin )\), we write \[ \Hom _{\Oo ^{\otimes }}^{\alpha }(X,Y) \qquad := \qquad \fib _{\alpha }(\, p_{\Oo }\colon \Hom _{\Oo ^{\otimes }}(X,Y) \to \Hom _{\Span (\Fin )}(I,J) \, ) \] for all \(X,Y \in \Oo ^{\otimes }\) with \(p_{\Oo }(X) \cong I\) and \(p_{\Oo }(Y) \cong J\).