Yes, the prerequisites of multiclassing are taken from both the class you are going to take and the class you already have.
Prerequisites
To qualify for a new class, you must meet the ability score prerequisites for both your current class and your new one, as shown in the Multiclassing Prerequisites table. For example, a barbarian who decides to multiclass into the druid class must have both Strength and Wisdom scores of 13 or higher. Without the full training that a beginning character receives, you must be a quick study in your new class, having a natural aptitude that is reflected by higher-than-average ability scores.
I tried building a Dexterity Paladin and multiclass to Warlock but when I try to add Warlock, even though my Charisma is 17, it requires Strength 13.
Only when I change from Dexterity 15 to Strength 15 can I add the Warlock (and then change it back).
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Yes, the prerequisites of multiclassing are taken from both the class you are going to take and the class you already have.
I have a similar issue when mutliclassing from warlock into paladin
in the PHB it states to multiclass into Paladin I need 13 Str and 13 Cha
My warlock has 14 Str and 18 Cha but when I try to add Paladin to my class it says that I need 15 Str to multiclass into Paladin.
Seeing as the PHB has been out far longer then dndbeyond I think something is off with the character multiclass options in the character creator
I’ll try it again maybe I missed something but it was unclickable for me and said I needed 15 str to multiclass into it