Let $\Gamma$ be defined where objects are all finite sets and whose morphisms from $S$ to $T$ are maps $ \theta:S\rightarrow P(T)$ where if $\alpha,\beta\in S$ such that $\alpha\neq\beta$, then $\theta(\alpha)\cap\theta(\beta)=\emptyset$. The composite of $\theta:S\to P(T)$ and $\phi:T\to P(U) $ is $\psi: S\to P(U)$ where $\psi(\alpha)=\cup_{\beta\in\theta(\alpha)}\phi(\beta)$ for all $\alpha\in S$. We have that $\Gamma$ is a category. Let $Fin_\star$ denote the category whose objects are all finite pointed sets and whose morphisms are base-point-preserving functions. Then $Fin_\star^{op}\simeq \Gamma$.
I proved that $\Gamma$ is a category. Now I need help defining functors $F:Fin_\star^{op} \to \Gamma$ and $G:\Gamma\to Fin_\star^{op}$ along with natural isomorphisms $\alpha:id_{Fin_\star^{op}}\simeq GF$ and $\beta:FG\simeq id_\Gamma$.
Try the functor $G: \Gamma \rightarrow Fin_*^{op}$ that sends a finite set $S$ to $S \cup \{\ast \}$ where $\ast$ is the point. Then we can send the function $\theta: T \rightarrow P(S)$ to the morphism of pointed sets $S \cup \{\ast \} \rightarrow T \cup \{\ast \}$ is such that the preimage of each $t \in T$ is $\theta(t)$ and everything else is sent to $\ast$. (This works because of the disjointness condition in the definition of morphisms of $\Gamma$).
You should check that this is a functor. Then, you can either use the idea to work out what $F: Fin_*^{op} \rightarrow \Gamma$ should be, or (probably easier) prove that it is full, faithful and essentially surjective.