If a cleric takes the Ritual Caster feat and picks the wizard list, are they able to scribe rituals that are on both classes’ spell lists?

I am playing a cleric. I would like to take the Ritual Caster feat at level 4 to access some wizard spells for utility.

Can my cleric spend the material cost to add rituals shared by both classes (like Detect Magic, Detect Poison, Identify, Gentle Repose, etc.) to the ritual book? Or do I need to find a wizard based written source?