solifine

To write the operadic multiplications γ: [r] ⨿ ([m₁] ⨿ ⋯ ⨿ [mᵣ]) → [Σmᵢ], maybe it's good to write elements like (k,r) and (k,mᵢ) to specify which component we're starting in. ⌊05⌋

(ofc you should be careful, bc in real life you could have, like, m₁=m₂=r=5 and then you'll get confused) ⌊06⌋

Are you satisfied that the standard simplices form an operad? If so, let X be a simplicial set. We know that hom(A⨿B,X) = hom(A,X)×hom(B,X) and X(n) = hom(Δⁿ,X). (here, hom means "simplicial set morphisms", while earlier it meant morphisms in 𝚫) ⌊09⌋

It follows that the maps γ from above give functions X(Σmᵢ) → X(r)×(X(m₁)×⋯×X(mᵣ)) and the unique map ∅ → [1] gives X(1) → hom(∅,X) = ∗. ⌊10⌋

Because the maps γ were not injective, the cocomposition map X(Σmᵢ) → X(r)×(X(m₁)×⋯×X(mᵣ)) lands in a smaller subset. ⌊11⌋

More specifically, this is an X(1)-colored cooperad. The "output color" of an element x ∊ X(n) is found using the endpoint preserving function [1] → [n] in 𝚫 (that is, 0↦0, 1↦n). ⌊12⌋