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?