A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any nice description of the corresponding categories of algebras and coalgebras?
If I am not mistaken, when $A$ is a finitely generated projective $k$-module, $\operatorname{Hom}_k(A,A\otimes_k-)$ is isomorphic to $\operatorname{Hom}_k(A,A)\otimes_k-$, so that the category of $T_A$-algebras is equivalent to the category of $\operatorname{Hom}_k(A,A)$-modules.
Again if I am not mistaken, when $A$ is the quotient of $k$ by an ideal $I$, the monad is idempotent, and we get the category of $k/I$-modules.
One more thing I figured out is that the adjunction induces a comparison functor from $k$-modules to $T_A$-algebras; that is, modules of the form $\operatorname{Hom}_k(A,X)$ have structures of $T_A$-algebras, in a functorial way. Similarly, modules of the form $A\otimes_kX$ have structures of $C_A$-coalgebras. But I could not figure out how to use this.
Simplest example where I got stuck: when $k$ is the algebra of univariate polynomials over some field. Thus $A$ is given by a linear operator $f:V\to V$. What are the $T$-algebras and $C$-coalgebras in this case?
There are obvious analogs of this question for other monoidal closed categories, for example for an object in a cartesian closed category $\bf C$. In the latter case, if I am not mistaken, the category of coalgebras for the comonad $A\times(-)^A$ is equivalent to the category of internal functors from the antidiscrete groupoid on $A$ (with the object of objects $A$ and object of morphisms $A\times A$) to $\bf C$. I could not however find a good description of algebras for the monad $(A\times-)^A$.
But let us stick to the case of modules here.
