I want to show that $\textbf{HP}_\kappa$ implies $\textbf{HK}_\kappa$, Prikry hypothesis and Kurepa hypothesis in $\kappa$, respectively. Where \begin{equation} \textbf{HK}_\kappa: \text{ There's a family }\mathcal{F}\subset\mathcal{P}(\kappa)\text{ such that, }|\mathcal{F}|\geq\kappa^+\text{, and } \forall\alpha<\kappa |\{\alpha\cap A:A\in\mathcal{F}\}|<\kappa. \end{equation} And \begin{equation} \textbf{HP}_\kappa: \text{ There's a family } \mathcal{F} \subset \kappa^\kappa \text{ such that, } \forall\alpha(\omega\leq\alpha<\kappa \to |\{f\cap (\alpha\times\alpha):f\in\mathcal{F}\}|<|\alpha|)\text{ and for any } g\in\kappa^\kappa\text{ exists }f\in\mathcal{F}\text{ such that } |\{\alpha<\kappa : f(\alpha)\leq g(\alpha)\}|<\kappa. \end{equation}
The definition of $\mathcal{F}$ for $\textbf{HK}_\kappa$ is more or less trivial; and it satisfies $\forall\alpha<\kappa |\{\alpha\cap A : A \in \mathcal{F}\}|<\kappa$. I just want to prove that the cardinality is in fact $\geq\kappa^+$. I know i have to use a Cantor-like argument: we suposse it isn't truth, then we give an enumeration of $\mathcal{F}$, $\langle f_\alpha:\alpha<\kappa\rangle$, then construct some $h$ such that $\forall\alpha<\kappa( h\neq f_\alpha)$, but $|\{\beta<\kappa:f_\alpha(\beta)\leq h(\beta)\}|=\kappa$. But I'm struggling to construct the $h$.
Suppose that $F=\{f_\xi:\xi<\kappa\}\subseteq{^\kappa\kappa}$. For each $\eta<\kappa$ let
$$g(\eta)=\sup\{f_\xi(\eta)+1:\xi\le\eta\}\;;$$
then $g(\eta)>f_\xi(\eta)$ for each $\xi\le\eta$. Equivalently, for each $\xi<\kappa$ we have $f_\xi(\eta)<g(\eta)$ whenever $\xi\le\eta<\kappa$, so that
$$|\{\eta<\kappa:g(\eta)\le f_\xi(\eta)\}|<\kappa\;.$$
Thus, no family of fewer than $\kappa^+$ functions can satisfy the conclusion of $\mathbf{HP}_\kappa$.