Bsrhrki

Let $(mathbfM,otimes,1)$ be a closed monoidal category and $(mathbfC,oplus,0)$ an $mathbfM$-enriched monoidal category. Furthermore, assume that we have a copowering $odot:mathbfMtimesmathbfCto mathbfC$. Is there a canonical morphism
$$(Aodot X)oplus (Bodot Y)to (Aotimes B)odot (Xoplus Y)$$
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 $mathcalO$ in the above setting, we need for the associativity axiom a morphism
$$mathcalO(r)odot left(bigoplus_i (mathcalO(k_i)odot X^oplus k_i)right)toleft(mathcalO(r)otimesbigotimes_imathcalO(k_i)right)odot left(bigoplus_iX^oplus k_iright)$$
Or the other direction. If $mathbfM$ is considered to be enriched over itself, everything is fine because then $otimes=odot=oplus$, but in general?

No. Consider the case where $(M,otimes,1)$ is $(mathbfSet,times,1)$, so the enrichment is vacuous, and $(C,oplus,0)$ is $(mathbfSet,+,0)$, 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 $A = X = Y = 1$, $B = 0$.

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 $(A otimes B) odot (X oplus Y) to (A odot X) oplus (B odot Y)$.