Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $\odot:\mathbf{M}\times\mathbf{C}\to \mathbf{C}$. Is there a canonical morphism
$$(A\odot X)\oplus (B\odot Y)\to (A\otimes B)\odot (X\oplus Y)$$
The question came to my mind because in order to spell out the axioms (in one of the definitions) for an algebra over an operad $\mathcal{O}$ in the above setting, we need for the associativity axiom a morphism
$$\mathcal{O}(r)\odot \left(\bigoplus\_i (\mathcal{O}(k\_i)\odot X^{\oplus k\_i})\right)\to\left(\mathcal{O}(r)\otimes\bigotimes\_{i}\mathcal{O}(k\_i)\right)\odot \left(\bigoplus\_iX^{\oplus k\_i}\right)$$
Or the other direction. If $\mathbf{M}$ is considered to be enriched over itself, everything is fine because then $\otimes=\odot=\oplus$, but in general?



Two monoidal structures and copowering
**No.** Consider the case where $(M,\otimes,1)$ is $(\mathbf{Set},\times,1)$, so the enrichment is vacuous, and $(C,\oplus,0)$ is $(\mathbf{Set},+,0)$, with copowering $\odot$ given by $\times$.


Then the morphism you ask for would give a map
$$(A \times X) + (B \times Y) \longrightarrow (A \times B) \times (X + Y) $$


which doesn’t exist in general: consider $A = X = Y = 1$, $B = 0$.




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However, there is a natural map in the other direction. There are natural maps $A \to C(X,A \odot X)$ and $B \to C(Y,B \odot Y)$, the structure maps of the copowering. Also, the definition of enriched monoidal category includes the condition that $\oplus$ is an enriched bifunctor, so there’s a general map $C(X,X') \otimes C(Y,Y') \to C(X \oplus Y, X' \oplus Y')$. Putting these together, we get a map
$$ A \otimes B \longrightarrow C(X, A \odot X) \otimes C(Y, B \odot Y) \longrightarrow C(X \oplus Y, (A \odot X) \oplus (B \odot Y)) $$
which corresponds under copowering to a map $(A \otimes B) \odot (X \oplus Y) \to (A \odot X) \oplus (B \odot Y)$.


