Names for some subcategories

Take a category $$mathcal{C}$$ and a subcategory $$mathcal{S}$$, such that the objects of $$mathcal{S}$$ are the same as the objects of $$mathcal{C}$$, but $$mathcal{S}$$ is not a full subcategory of $$mathcal{A}$$. What is the name of such a subcategory?

Next: Assume that for any morphism $$f:x to y$$ in $$mathcal{S}$$ admitting a splitting $$s:y to x$$ in $$mathcal{A}$$, we automatically have that $$s$$ is a morphism in $$mathcal{S}$$. What is the name for such a subcategory?