(co)algebras of the adjunction between presheaves and bundles

Question

For a topological space $X$ and the lattice of open sets $LX$, there is an extension of the "inclusion" (don't know what to call it) functor $F:LX\to\text{Top}/X$ along the Yoneda embedding $y:LX\to\text{Set}^{LX^o}$, $\Lambda:\text{Set}^{LX^o}\to\text{Top}/X$.

In the other direction, a bundle $E$ is sent to the presheaf $\Gamma E=\text{Top}/X(F-,E)$. These two functors are an adjunction $\Lambda\dashv\Gamma$.

I want to understand the (co)algebras of the (co)monad. Are these equivalent to the full subcategories on which the adjunction is an equivalence?

Answer 1

The left adjoint $\Lambda$ is fully faithful, so the monad $\Gamma \Lambda$ is idempotent and the algebras are simply the presheaves $P$ such that the unit $\eta_P : P \to \Gamma \Lambda P$ is an isomorphism, i.e. sheaves on $X$.

Since the monad $\Gamma \Lambda$ is idempotent, the comonad $\Lambda \Gamma$ is also idempotent, and the coalgebras are simply the bundles $E$ such that the counit $\epsilon : \Lambda \Gamma E \to E$ is an isomorphism, i.e. the espaces étalés over $B$.