[05XE]
Proof. By 2.2.9 ∞ \infty
End 𝒞 red ( X ) \textstyle{\operatorname{End}_{\mathcal{C}}^{\operatorname{\scriptsize{red}}}\left(X\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝒞 \textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝔼 ∞ \textstyle{\mathbb{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝒞 ¯ ⊔ , \textstyle{\underline{\mathcal{C}}_{\sqcup},}
which, by 2.2.11
End 𝒞 red ( X ) ( m ) \textstyle{\operatorname{End}_{\mathcal{C}}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(m\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Mul 𝒞 ( X ( m ) , X ) \textstyle{\operatorname{Mul}_{\mathcal{C}}\left(X^{\left(m\right)},X\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝔼 ∞ ( m ) \textstyle{\mathbb{E}_{\infty}\left(m\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Mul 𝒞 ¯ ⊔ ( X ( m ) , X ) . \textstyle{\operatorname{Mul}_{\underline{\mathcal{C}}_{\sqcup}}\left(X^{\left(m\right)},X\right).}
The bottom map is the map Δ 0 → Map 𝒞 ( X ⊔ m , X ) \Delta^{0}\to\operatorname{Map}_{\mathcal{C}}\left(X^{\sqcup m},X\right) 𝔼 ∞ = ( Δ 0 ) ⊔ → 𝒞 ¯ ⊔ \mathbb{E}_{\infty}=\left(\Delta^{0}\right)_{\sqcup}\to\underline{\mathcal{C}}_{\sqcup} σ : X ⊔ m → X ⊗ m \sigma\colon X^{\sqcup m}\to X^{\otimes m}
( − ) ⊔ : 𝐂𝐚𝐭 ∞ ⇆ 𝐎𝐩 ∞ un : ( − ) ¯ . \left(-\right)_{\sqcup}\colon\mathbf{Cat}_{\infty}\leftrightarrows\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\colon\underline{\left(-\right)}.
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