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Hi. There’s an MO question What are the internal categories in an endofunctor category which may be of interest.
A monad in a bicategory $K$ is a lax functor $1 \to K$. This definition admits a natural generalisation to a set $X$ equipped with a lax functor from the chaotic bicategory on $X_{\mathrm{ch}}}X$ (i.e. having a single 1-cell between each pair of elements of $X$), which is exactly a category enriched in $K$. Thus categories enriched in a bicategory are many-object generalisations of monads. They were called “polyads” by Bénabou. In particular, your $End(C)$-enriched categories are bicategory-enriched categories where every object has the same extent (namely, $C$). The paper Enriched categories as a free cocompletion by Garner and Shulman takes this perspective on enriched categories, for instance.
There’s also work by Orchard et al. here on category-graded monads:
Our approach can be summarised as the horizontal categorification of monads.
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