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Do monads which are monoidal and opmonoidal have a name? (Bimonoidal?) In case they have already been studied, who can point me to a reference?

More in detail. Let $(C,\otimes)$ be a symmetric (or braided) monoidal category. Let $(T,\eta,\mu)$ be a monad on $C$ such that:

  1. $T:C\to C$ is a bilax monoidal functor (compatible lax and colax monoidal structure);
  2. $\eta,\mu$ are monoidal natural transformations (commute with unit and multiplication);
  3. $\eta,\mu$ are also op-monoidal natural transformation (commute with the counit and comultiplication).

In other words, $T$ is a monad in the 2-category of categories, bilax monoidal functors, and (bi)monoidal natural transformations (meaning monoidal and opmonoidal).

What is the standard terminology? What would be a reference?

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