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I'm going over what's already in the comments a little here, but: given an action of $C$ on $D$ as described, any monoid in $C$ gives rise to a monad on $D$. This is because a monoid in $C$ is just a …

This is close to the subject of my (so far unfinished) thesis, so I'll try to explain for the benefit of future readers. We define an oplax action of a monoidal category $\mathcal X$ upon a categor …

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid in $[\mathcal C, \mat …

Let $\mathcal C,\mathcal D,\mathcal E$ be monoidal categories, let $g$ be an oplax monoidal functor from $\mathcal C$ to $\mathcal D$ and let $G$ be a lax monoidal functor from $\mathcal D$ to $\mathc …

$\newcommand{\C}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\F}{\mathcal F} \renewcommand{\H}{\mathcal H} \newcommand{\from}{\colon} \newcommand{\tensor}{\otimes} \require{AMScd}$ Given mono …