The bidualizing monad 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 $X$).

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*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 $\mu$.)
 A: 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

A. Kock, On double dualization monads, Math. Scand. 27 (1970), 151-165, pdf

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

E. Manes, Monads of sets, Handbook of algebra. Vol. 3. North-Holland, 2003, 67-153, link

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

F.E.J. Linton, Applied functorial semantics I, Annali di Matematica Pura ed Applicata 86 (1970), 1–14, pdf

The double dualization monad for $\mathbf{Vect}_k$ and the object $k$ has been discussed at MO/104777.
A: This is called the double dual monad, and has been studied since (at least) the 70’s. It’s often looked at in categorical treatments of functional analysis-type things.
Here is a paper by Kock:
https://www.mscand.dk/article/download/10995/9016
And a more recent paper by Lucyshyn-Wright:
http://www.tac.mta.ca/tac/volumes/29/31/29-31.pdf
