Let $X$ be a topological space, such that $(X, \mathcal{O}_X)$ is locally ringed. Let $A$ be a ring. We showed in the lecture that there is a natural bijection of $Hom((X, \mathcal{O}_X),(Spec(A), \mathcal{O}_{Spec(A)}))$ with $Hom(A,\mathcal{O}_{X}(X))$.
Now in the lecture there was a statement which I didn't understand. Every locally ringed space $X$ such that $\mathcal{O}_X(U)$ has characteristic $p > 0$ for all $U \subset X$ open admits a unique morphism to Spec($\mathbb{F}_p$). Conversely, if $X$ admits a morphism to Spec$(\mathbb{F}_p)$, then every $\mathcal{O}_X(U)$ has characteristic $p > 0$.
Now I know that the existence of a ring homomorphism $f:A \to B$ means $char(B) | char(A)$.
Edit: Idea: If we set $A:=\mathbb{F}_p$, then do we find (since $char(\mathcal{O}_X(X))= p$) a morphism from $\mathbb{F}_p \to \mathcal{O}_X(X)$ ? Because then we could use the Hom-bijection I think.
The main point is that if $R$ is any ring admitting a ring homomorphism of the form $\mathbb{F}_p \to R$ for some $p$, then this morphism is unique: this is because $\mathbb{Z}$ admits exactly one morphism to $R$ (since $1 \in \mathbb{Z}$ must be sent to $1 \in R$) and the morphism $\mathbb{F}_p \to R$ has to be induced by this one by the same logic.
Uniqueness of the maps $\mathbb{F}_p \to \mathcal{O}_X(U)$ for every $U \subseteq X$ show that the induced morphisms of locally ringed spaces $(U, \mathcal{O}_X \mid_U) \to (\text{Spec}(\mathbb{F}_p), \mathcal{O}_{\text{Spec}(\mathbb{F}_p)})$ glue together, and provide a well-defined unique morphism of locally ringed spaces $(X, \mathcal{O}_X) \to (\text{Spec}(\mathbb{F}_p), \mathcal{O}_{\text{Spec}(\mathbb{F}_p)})$ (recall that morphisms of sheaves glue!)