Fix a monoidal category $(M,\otimes,I)$ and a small discrete monoidal subcategory $M'\subseteq M$. Define a new monoidal category $C:=CoKl(M,M')$ as follows: $Ob(C):=Ob(M)$, and for any $X,Y\in Ob(C)$, $$Hom_C(X,Y):=\sum_{m\in M'} Hom_M(X\otimes m,Y).$$ There is thus a projection $Hom_C\to M'$. The identity morphisms project to the unit, $m=I$ and use the identity morphisms from $M$. Given morphisms $\phi\colon X\otimes m_1\to Y$ and $\psi\colon Y\otimes m_2\to Z$ the composition $\psi\circ\phi$ projects to $m_1\otimes m_2$ and is given in the obvious way by $$X\otimes (m_1\otimes m_2)\to Y\otimes m_2\to Z.$$

Now doesn't $C$ look like the cokleisli category of some comonad? But of course it's not, in general. There's a functor $M\to C$, but not generally an adjoint.

If $M=\pi_0(M)$ is a discrete monoidal category, then this construction gives something like the quotient, namely $\pi_0CoKl(M,M')\cong M/M'.$

Is there a name or reference for this coKleisli-like construction?

Fix a symmetric monoidal category $(M,\otimes,I)$ and a small discrete monoidal subcategory $M'\subseteq M$. Define a new symmetric monoidal category $C:=CoKl(M,M')$ as follows: $Ob(C):=Ob(M)$, and for any $X,Y\in Ob(C)$, $$Hom_C(X,Y):=\sum_{m\in M'} Hom_M(X\otimes m,Y).$$ There is thus a projection $Hom_C\to M'$. The identity morphisms project to the unit, $m=I$ and use the identity morphisms from $M$. Given morphisms $\phi\colon X\otimes m_1\to Y$ and $\psi\colon Y\otimes m_2\to Z$ the composition $\psi\circ\phi$ projects to $m_1\otimes m_2$ and is given in the obvious way by $$X\otimes (m_1\otimes m_2)\to Y\otimes m_2\to Z.$$ The monoidal structure on $C$ is straightforward.

Now doesn't $C$ look like the cokleisli category of some comonad? But of course it's not, in general. There's a functor $M\to C$, but not generally an adjoint.

If $M=\pi_0(M)$ is a discrete monoidal category, then this construction gives something like the quotient, namely $\pi_0CoKl(M,M')\cong M/M'.$

Is there a name or reference for this coKleisli-like construction?