Let $\mathbf{C}$ be a closed symmetric monoidal category (I probably need even less than this; the examples I have in mind are simply the category of modules over a commutative ring and the category of sets) and let $\omega$ be an (arbitrary) object in $\mathbf{C}$ which I will term the “dualizing object”. Define a contravariant functor $D\colon \mathbf{C} \to \mathbf{C}$ taking $X$ to $[X,\omega]$ (internal Hom) and $X\to Y$ to the composition map $[Y,\omega] \to [X,\omega]$: let us call $DX$ the “dual” of $X$.


Now let $T = D^2$ be the covariant functor taking an object to its “bidual”. Call $\eta\colon 1\_{\mathbf{C}}\to T$ the natural transformation $\eta\_X \colon X \to D^2X = [[X,\omega], \omega]$ obtained from the evaluation map $X \otimes [X,\omega] \to \omega$. And define a natural transformation $\mu\colon T^2 \to T$ by letting $\mu\_X \colon D^4 X \to D^2 X$ be $D(\eta\_{DX})$.


**Fact:** $(T,\eta,\mu)$ is a monad.


This is probably a well-known observation, and it is certainly not difficult (although it is tedious to check, or at least I found it tedious, having to go up to the sexiesdual(!) $D^6X$ of $X$).


* Does this monad have a name? (The “bidual monad” perhaps?) Is there a standard reference for it?
* What are some “natural” occurrences, if any, of algebras for this monad?


(This came up to me by asking myself whether the sequence of iterated biduals $T^n X$ of an object stabilizes: the fact that $T$ is a monad says that, somehow, even though $T^2$ and $T$ are not the same, there is still a form of idempotency to $T$ in the existence of $\mu$.)



The bidualizing monad
It has been explained by Maxime Ramzi in the comments that this monad simply arises from the adjunction $[-,\omega] \vdash [-,\omega]^{\mathrm{op}}$.


As for the name, it's called the double dualization monad. The classical reference is



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> A. Kock, *On double dualization monads*, Math. Scand. 27 (1970), 151-165, [pdf](https://www.mscand.dk/article/download/10995/9016)
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The double dualization monad classifies algebra structures on a given object, see Theorem 3.2 in Kock's paper. The special case for $\mathbf{Set}$ appeas as Proposition 3.14 in



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> E. Manes, *Monads of sets*, Handbook of algebra. Vol. 3. North-Holland, 2003, 67-153, [link](https://www.sciencedirect.com/science/article/pii/S1570795403800591)
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The double dualization monad also appears in Linton's "contravariant representation theorem", see for example Theorem 3.53 in Manes' article. The classical reference for this is



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> F.E.J. Linton, *Applied functorial semantics I*, Annali di Matematica Pura ed Applicata 86 (1970), 1–14, [pdf](https://link.springer.com/content/pdf/10.1007/BF02415703.pdf)
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The double dualization monad for $\mathbf{Vect}\_k$ and the object $k$ has been discussed at [MO/104777](https://mathoverflow.net/questions/104777/what-are-the-algebras-for-the-double-dualization-monad).


