Here's the formulation for group theory; replace "group" by whatever to get it in other categories (it usually holds whenever it may be formulated, except you usually don't need to say "normal").
Let G be a group, and let H be a subgroup of G and N a normal subgroup of G such that G=HN. Denote the intersection of H and N by L=H∩N. Then L is a normal subgroup of H, and H/L is isomorphic to G/N.
This is the diagram to bear in mind:
the two "normal" quotient groups (along the "/" lines) are isomorphic!