Let $2$ be the discrete topological space with two elements. For a map of sets $$\beta : X \times Y \rightarrow 2 $$ We get a topology on $X$ and a topology on $Y$. The topology on $X$ is the weakest topology such that the map $\beta_y : X \rightarrow 2$ sending $x$ to $\beta(x, y)$ is continuous for each $y$. Call a topological space arising in this way a weak topology.
For a category $C$, write $[X, Y]_C$ for the hom-class of maps from an object $X$ to an object $Y$.
Under this definition, stone duality gives a functor $F : \text{Top} \rightarrow \mathbb{F}_2 \text{-alg}$ sending a topological space $X$ to $[X, 2]_{\text{Top}}$. There is a functor $\mathbb{F}_2 \text{-alg} \rightarrow \text{Top}$ sending an $\mathbb{F}_2$-algebra $A$ to $[A, \mathbb{F}_2]_{\mathbb{F}_2}$, where we put a weak topology on $[A, \mathbb{F}_2]_{\mathbb{F}_2}$ induced by the pairing $$ A \times [A, \mathbb{F}_2]_{\mathbb{F}_2} \rightarrow 2 $$ Sending $(a, \phi)$ to $\phi(a)$. $F$ and $G$ are adjoint to each other, and factor through a categorical equivalence $\text{StoneSpace} \leftrightarrow \mathbb{F}_2 \text{-alg}$, where $\text{StoneSpace}$ is the full subcategory of topological spaces consisting of stone spaces.
I am interested in the following modification:
Define a functor $\Phi : \text{Top} \rightarrow \text{Top-} \mathbb{F}_2 \text{-alg}$ by sending a topological space $X$ to $[X, 2]$, where we give $[X, 2]_{\text{Top}}$ the topology induced by the pairing $$ X \times [X, 2] \rightarrow 2 $$ sending $(x, \phi)$ to $\phi(x)$. Define a functor $\Psi : \text{Top-} \mathbb{F}_2 \text{-alg} \rightarrow \text{Top}$ by sending a topological boolean algebra $A$ to $[A, \mathbb{F}_2]_{\text{Top-} \mathbb{F}_2 \text{-alg}}$, where $\mathbb{F}_2$ gets the discrete topology. The topology on $[A, \mathbb{F}_2]_{\text{Top-} \mathbb{F}_2 \text{-alg}}$ is induced by the pairing $$ [A, \mathbb{F}_2]_{\text{Top-} \mathbb{F}_2 \text{-alg}} \times A \rightarrow 2 $$ sending $(\hat{x}, a)$ to $\hat{x}(a)$.
I can show abstractly that $\Phi$ and $\Psi$ factor through a categorical equivalence $C' \leftrightarrow D'$, where $C'$ is a full subcategory of $\text{Top}$ and $D'$ is a full subcategory of $\text{Top-} \mathbb{F}_2 \text{-alg}$.
Question: Can anyone concretely identify this category $C'$?
Notes:
1) Both mentioned adjunctions are idempotent
2) $C'$ contains profinite sets.