third isomorphism theorem

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