An article by Kennison
On limit preserving functors, Illinois J. Math. Volume 12, Issue 4 (1968), 616-619.
States the following
Th. Let $C$ be a small category. Then $[C,\text{Set}]_{\Gamma}$ is a reflective subcategory of $[C,\text{Set}]$.
where by $[C,\text{Set}]_{\Gamma}$ he means those functors preserving $\Gamma$-limits for $\Gamma$ a class of cones.
Is it possible to dualize this theorem in the following way:
Conj. Let $C$ be a small category. Then $[C,\text{Set}]^{\Gamma}$ is a coreflective subcategory of $[C,\text{Set}]$.
where by $[C,\text{Set}]^{\Gamma}$ I mean those functors preserving $\Gamma$-colimits for $\Gamma$ a class of cocones.
But is this dualization sound?
In this case $[C,\text{Set}]^{\Gamma}$ would be the category of fixed points of a suitable comonad $T: [C,\text{Set}] \to [C,\text{Set}]$. (A fixed point of $T$ is an object such that $T(X)\cong X$ via the counit of the comonad.)
I am even more interested in knowing if the following is true.
Hope. Is the $[C,\text{Set}]^{\Gamma}$ the category of the fixed points of a pointed (not copointed!) endofunctor $T: [C,\text{Set}] \to [C,\text{Set}]?$