In the definition of the category of smooth manifolds it's usual to take the morphisms to be simply smooth functions, and this results in having monomorfisms and epimorphisms to be injective and surjective smooth maps.
What would the morphisms need to be, with smooth manifolds as objects, to make the monos and epis to be immersions and submersions and keep diffeomorphisms as the isomorphisms? Is there a way to do this? Would that make the subobjects to be submanifolds?