Pointed evaluation equivalence

Proof. On underlying \(\infty \)-categories, evaluation identifies the category of arrows from the zero object with \(\Oo _{\lra {1}}\). It remains to prove full faithfulness on multimorphism animae.

Let \(\{F_i\}_{i\in I}\) and \(G\) be colors of \(\Oo _*\). By Theorem 18.3.5, their multimorphism anima is an end over \([1]^I\). We collapse its variables one at a time. After applying Fubini, the variable to be collapsed appears as the end of a bifunctor \(T\colon [1]\catop \times [1]\to \An \), allowing arbitrary fixed spectator colors. This end is the pullback \[ T(0,0)\times _{T(0,1)}T(1,1). \] Both \(T(0,0)\) and \(T(0,1)\) are contractible because \(F_i(0)\) is the zero object and hence operadic initial. Thus this pullback is equivalent to \(T(1,1)\). Fubini for ends and induction, with the values \(F_i(1)\) already obtained treated as spectator colors, identify the original end with \[ \Oo (\{F_i(1)\}_{i\in I};G(1)). \] By naturality of the end formula, this equivalence is the map induced by evaluation at \(1\). □