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The nlab page says

A (Set-based) operad is a monoid in the monoidal category $(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.

The monoidal structure is given by the so called complicated substitution product $∘$, which is roughly defined as follows.

The cardinal sum is a functor $ℙ\times ℙ\to ℙ$, it extends (co-continuously) to a tensor product on $Psh(ℙ)$ by the Day Convolution. Then one argues that $$\underline{\hom}(Psh(ℙ), Psh(ℙ))=Psh(ℙ)$$ where $\underline{\hom}$ is taking respect to the above tensor product. Then we transfer the obvious composition on $\underline{\hom}(Psh(ℙ), Psh(ℙ))$ to $Psh(ℙ)$, obtaining the substitution product.

My question is that whether the substitution product itself arises from some functor $ℙ\times ℙ\to ℙ$ via the Day convolution?

It seems we can define a substitution product on $ℙ$ by the genuine substitution operations.

The nlab page says

A (Set-based) operad is a monoid in the monoidal category $(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.

The monoidal structure is given by the so called complicated substitution product $∘$, which is roughly defined as follows.

The cardinal sum is a functor $ℙ\times ℙ\to ℙ$, it extends (co-continuously) to a tensor product on $Psh(ℙ)$ by the Day Convolution. Then one argues that $$\underline{\hom}(Psh(ℙ), Psh(ℙ))=Psh(ℙ)$$ where $\underline{\hom}$ is taking respect to the above tensor product. Then we transfer the obvious composition on $\underline{\hom}(Psh(ℙ), Psh(ℙ))$ to $Psh(ℙ)$, obtaining the substitution product.

My question is that whether the substitution product itself arises from some functor $ℙ\times ℙ\to ℙ$ via the Day convolution?

It seems we can define a substitution product on $ℙ$ by the genuine substitution operations.

Added. As pointed below, the substitution product is not bi-co-continuous. Now I should ask whether the substitution product restricts to representables, if yes, then how it extends to the $Psh(ℙ)$.

The aim of my question is to understand the substitution product on the smaller space $ℙ$ as much as possible.

The nlab page says

A (Set-based) operad is a monoid in the monoidal category $(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.

The monoidal structure is given by the so called complicated substitution product $∘$, which is roughly defined as follows.

The cardinal sum is a functor $ℙ\times ℙ\to ℙ$, it extends (co-continuously) to a tensor product on $Psh(ℙ)$ by the Day Convolution. Then one argues that $$\underline{\hom}(Psh(ℙ), Psh(ℙ))=Psh(ℙ)$$ where $\underline{\hom}$ is taking respect to the above tensor product. Then we transfer the obvious composition on $\hom(Psh(ℙ), Psh(ℙ))$ to $Psh(ℙ)$, obtaining the substitution product.

My question is that whether the substitution product itself arises from some functor $ℙ\times ℙ\to ℙ$ via the Day convolution?

It seems we can define a substitution product on $ℙ$ by the genuine substitution operations.

The nlab page says

A (Set-based) operad is a monoid in the monoidal category $(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.

The monoidal structure is given by the so called complicated substitution product $∘$, which is roughly defined as follows.

The cardinal sum is a functor $ℙ\times ℙ\to ℙ$, it extends (co-continuously) to a tensor product on $Psh(ℙ)$ by the Day Convolution. Then one argues that $$\underline{\hom}(Psh(ℙ), Psh(ℙ))=Psh(ℙ)$$ where $\underline{\hom}$ is taking respect to the above tensor product. Then we transfer the obvious composition on $\underline{\hom}(Psh(ℙ), Psh(ℙ))$ to $Psh(ℙ)$, obtaining the substitution product.

My question is that whether the substitution product itself arises from some functor $ℙ\times ℙ\to ℙ$ via the Day convolution?

It seems we can define a substitution product on $ℙ$ by the genuine substitution operations.

The nlab page says

A (Set-based) operad is a monoid in the monoidal category $(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.

The monoidal structure is given by the so called complicated substitution product $∘$, which are basically defined as follows.

The cardinal sum is a functor $ℙ\times ℙ\to ℙ$, it extends (co-continuously) to a tensor product on $Psh(ℙ)$ by the Day Convolution. Then one argues that $$\hom(Psh(ℙ), Psh(ℙ))=Psh(ℙ)$$ where $\hom$ is respect to the above tensor product. Then we transfer the obvious composition on $\hom(Psh(ℙ), Psh(ℙ))$ to $Psh(ℙ)$, obtaining the substitution product.

My question is that whether the substitution product itself arises from some functor $ℙ\times ℙ\to ℙ$ via the Day convolution?

It seems we can define a substitution product on $ℙ$ by the genuine substitution operations.

The nlab page says

A (Set-based) operad is a monoid in the monoidal category $(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.

The monoidal structure is given by the so called complicated substitution product $∘$, which is roughly defined as follows.

The cardinal sum is a functor $ℙ\times ℙ\to ℙ$, it extends (co-continuously) to a tensor product on $Psh(ℙ)$ by the Day Convolution. Then one argues that $$\underline{\hom}(Psh(ℙ), Psh(ℙ))=Psh(ℙ)$$ where $\underline{\hom}$ is taking respect to the above tensor product. Then we transfer the obvious composition on $\hom(Psh(ℙ), Psh(ℙ))$ to $Psh(ℙ)$, obtaining the substitution product.

My question is that whether the substitution product itself arises from some functor $ℙ\times ℙ\to ℙ$ via the Day convolution?

It seems we can define a substitution product on $ℙ$ by the genuine substitution operations.