Subfunctor from a category to Set

Question

I'm looking for the definition of a subfunctor $R$ of a functor $U:\cal K \to \mathbb {Set}$. Are there any other conditions beyond that $R(A)\subseteq U(A)$ for all objects $A\in \cal K$ ?

Answer 1

On top of the condition that $R(A)\subseteq U(A)$ for every object $A$ in the category $\mathcal{K}$, you need the morphisms of $\mathcal{K}$ to behave accordingly: If $f:A\to B$ is a morphism in $\mathcal{K}$, then $R(f)$ should send $R(A)$ to $R(B)$. Since $R$ is a subfunctor of $U$, $R(f)$ is the restriction of $U(f)$ to the set $R(A)$, we may write this condition as $$Uf[R(A)]\subseteq R(B).$$ Now remember that this should hold simultaneously for all the objects and morphisms in $\mathcal{K}$!