Some categories $C$ have a full subcategory $D$ that is simultaneously reflective and coreflective with the same functor $C \to D$ serving as both the reflector and the coreflector. For example, if $C$ is any category with a zero object $0$, then the full subcategory consisting of just the zero objects (which are all uniquely isomorphic to $0$) is simultaneously reflective and coreflective, and the left adjoint and right adjoint to the inclusion both map each object of $C$ to $0$.
Now, here are two questions:
Questions:
- Given an endofunctor $T:C \to C$ with natural transformations $\eta:1_C \Rightarrow T$ and $\epsilon:T \Rightarrow 1_C$ for which $\eta \circ \epsilon=1_T$, are $T\eta=\eta{T}$ and $T\epsilon=\epsilon{T}$ necessarily inverse natural isomorphisms? (For the former equation to hold, it suffices to require $\eta$ to have epic components, while for the latter, it suffices to require $\epsilon$ to have monic components.)
- With $T$, $\eta$, and $\epsilon$ as in Question 1, do $\eta$ and $\epsilon$ uniquely determine each other?
Attempt:
Since clearly, we have $T\eta \circ T\epsilon=1_{T^2}$, it suffices to show that we also have $T\epsilon \circ T\eta=1_T$.
But first, we must show that $T\eta=\eta{T}$ and $T\epsilon=\epsilon{T}$. Naturality of $\eta$ implies that $T\eta \circ \eta=\eta{T} \circ \eta$, hence $T\eta=\eta{T}$ since $\eta$ is a split epimorphism. Likewise, naturality of $\epsilon$ implies that $\epsilon \circ T\epsilon=\epsilon \circ \epsilon{T}$, hence $T\epsilon=\epsilon{T}$ since $\epsilon$ is a split monomorphism.
Now, by the last paragraph and naturality of $\epsilon$ (or $\eta$), we have $T\epsilon \circ T\eta=\epsilon{T} \circ T\eta=\eta \circ \epsilon=1_T$ (or $T\epsilon \circ T\eta=T\epsilon \circ \eta{T}=\eta \circ \epsilon=1_T$). $\square$
In particular, the full subcategory consisting of the objects $X$ for which $\eta_X$ (or equivalently, $\epsilon_X$) is an isomorphism is simultaneously reflective and coreflective with the same functor serving as both the reflector and the coreflector since $T$ is simultaneously an idempotent monad (with $\mu=T\epsilon=\epsilon{T}$) and an idempotent comonad (with $\delta=T\eta=\eta{T}$).
The above only answers Question 1. What I need help with is Question 2.
By the Eckmann-Hilton argument, the monoid $\operatorname{End}(1_C)$ is always commutative, but I'm not sure whether this is sufficient to answer Question 2.
Perhaps, more generally, if an object $X$ (e.g. the identity functor $1_C$ considered as an object of the category of endofunctors of $C$) of a category has a commutative endomorphism monoid, then for any object $Y$ and any two morphisms $f:X \to Y$ and $g:Y \to X$ for which $f \circ g=1_Y$, making $Y$ into a retract of $X$, the morphisms $f$ and $g$ uniquely determine each other. Even more generally, perhaps, this continues to hold if we merely require that any two idempotent endomorphisms of $X$ commute with each other.