Where did the idea of Pyramid Head is against the Entity came from, because it isn't true lorewise
Entity sought out Pyramid Head after Silent Hill
"When his duty was complete and his presence no longer needed, he prepared for the long rest — and yet, his skills were required elsewhere. The Fog that streamed over him was somehow different than that he was accustomed to in Silent Hill, as if each wisp contained the nerves of a creature, writhing, seeking him out.
There was an unspoken agreement in that moment. The billowing cloud was an invitation to duty and sadism, and Pyramid Head, taking a step into The Fog, accepted his obligation once more"
both are working in a symbiotic relation, as Pyramid Head came to the fog by his own will, the cages and final judgement mori is not against the Entity as they do the same job as hooks (you get same amount of points as hooks) and functions more like Amanda's bear traps, a unique way to sacrifice to the Entity