Multimorphisms Day operad

Proof. Let us write \(\Oo := \oDay (\Mm _C, \Mm _D)\) for simplicity. By definition, the anima of multimorphisms \(\Oo (\{F_i\}_{i \in I}; G)\) is the fiber of the projection map \[ p_{\Oo }\colon \Hom _{\Oo ^{\otimes }}(\{F_i\}_{i \in I}, G) \to \Hom _{\Span (\Fin )}(I, \lra {1}) \] over the active span \(I \xleftarrow {\;\id \;} I \to \lra {1}\). By Theorem 16.2.4, the operad \(\Oo \) sits in the following pullback square of \(\infty \)-operads:

Since hom animae in a pullback of \(\infty \)-categories form a pullback of animae, \(\Hom _{\Oo ^{\otimes }}(\{F_i\}_{i \in I}, G)\) is itself the fiber in a pullback square of hom animae. Taking the fibers of these hom animae over the active span \(I \to \lra {1}\), we find that \(\Oo (\{F_i\}_{i \in I}; G)\) is the fiber in the following pullback square of animae: \begin {equation*}

\end {equation*} Here the object \(\Un (F_i) \in \Ar ^{\oplax }\) is the cartesian unstraightening of the functor \([1]\catop \to \Cat _{\infty }\) which takes the arrow \(1 \to 0\) to \(F_i\colon C \to D\), and similarly for \(\Un (G)\). Since \(\Cat _{\infty }\) and \(\Ar ^{\oplax }\) have finite products, we may write the three multimorphism animae in the source and target of the right vertical map as hom animae in \(\Ar ^{\oplax }\) and \(\Cat _{\infty }\) out of the corresponding products, which results in a pullback square as follows:

By applying Theorem 16.2.2 to the trivial \(\infty \)-operad, we get that the functor \((s,t)\colon \Ar ^{\oplax } \to \Cat _{\infty } \times \Cat _{\infty }\) is a cartesian fibration over \(\Cat _{\infty } \times \{D\}\) and a cocartesian fibration over \(\{C\} \times \Cat _{\infty }\). The fibers of \((s,t)\) are given by the functor categories \(\Fun (C,D)\), with functoriality in \(C\) given by precomposition and functoriality in \(D\) given by postcomposition. The fiber of the upper-right hom anima over a pair \((f,g)\) is therefore the anima of natural transformations \(g\circ \prod _iF_i\to G\circ f\). In particular, Equation 16.1 identifies the required fiber as \[ \Oo (\{F_i\}_{i \in I}; G) \simeq \Hom _{\Fun (C^I,D)}(\bigotimes _D^I \circ \prod _{i \in I} F_i, G \circ \bigotimes _C^I), \] as claimed. □