A subfunctor of a set-valued functor assigns a subset to each object, preserved by every action of on a morphism. The inclusions form a natural transformation. Equivalently, it is a subobject of in the functor category.
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In category theory, a **subfunctor** is a concept that extends the idea of a subobject to the context of functors. While subobjects represent "parts" of objects in a category, subfunctors represent "parts" of functors in a more structured manner. ### Definition Let \( F: \mathcal{C} \to \mathcal{D} \) be a functor.