If a fighter multiclasses into sorcerer, when they cast spells do they also replace material components with somatic?

From the Sorcerer class page,

Because you’re a sorcerer, you can usually replace material components with somatic components, so you don’t need to use a material component pouch.

I interpret this to mean that whenever a sorcerer casts a spell that requires a material component (without a cost), then they can choose to replace that component with a somatic one. From the listing on spell components, this means that the spell would keep the manipulate trait, but one important thing would change: the spell no longer requires you to have a hand free to retrieve and manipulate the material component.

Now, if a fighter takes the sorcerer dedication, then from the dedication page,

You cast spells like a sorcerer.

I interpet this to mean that the fighter also gains the ability to replace any material component with a somatic component, and that as a consequence the fighter can cast sorcerer spells without needing a hand free.

Is my interpretation correct?