Is there a construction which maps each object in a category to the set of its subobjects?
Concretely, I'm interested in mapping an object $M$ in the category of manifolds $\mathbf{Man^1}$ to the set of its submanifolds $P(M)$.
It seems a morphism $f: A \rightarrow B$ would need to be mapped to $f \circ i: P(A) \rightarrow P(B)$, where $i$ is the injection map $i: A' \hookrightarrow A$ for a given submanifold $A'$. So in a sense, I'm interested restricting the domain and codomain of a function
This seems related to the concept of a well-powered category which is required of the starting category to make this construction, but the queries "well-powered functor" and "subobject functor" don't seem to yield something I can use here.