Duality for $\kappa$-additive complete atomic modal algebras

Let $\kappa$ be a cardinal number. The category $\mathbf{CAMA}_{\kappa}$ of $\kappa$-additive complete atomic modal algebras is dually equivalent to the category $\mathbf{MRKF}_{\kappa}$ of $\kappa$-downward directed multi-relational Kripke frames, since the category $\mathbf{NFr}_{\kappa}$ of $\kappa$-complete neighborhood frames is dually equivalent to $\mathbf{CAMA}_{\kappa}$ and is equivalent to $\mathbf{MRKF}_{\kappa}$. We present another direct proof of this duality for any regular cardinal $\kappa$.