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:

**G**
/ \
**N** **H**
\ /
**L**

the two "normal" quotient groups (along the "/" lines) are isomorphic!