Key things to remember...
-Law grew up in the "White City." Marijoies? The town Robin was teleported to and spent 2 years living?
-Doflamingo mother died at 8 and he killed his father at 10
-Law wants Doflamingo dead for killing Corazon
-When Law was remembering Corazon, the silouhette looked like Doflamingo but why have it so darkened? Was Doffy and Corazon twins?
-Doffy is a Tenryubitto but when he was drawn as a kid, he was wearing rags. He looked very poor
Doflamingo had a Tenryubitto father but his mother was woman who was forced to become the Tenryubitto's wife. The Tenryubitto father was very abusive and of course, neglected the twins Doffy and Corazon. One day, the mother was killed by the father and Doffy probably blamed Corazon. Alternatively, Doffy development a serious hatred of the world whilst Corazon still had hope and Doffy didn't like it. By the time they were 10, Doffy killed the father. A few years later, they left Marijoies to become pirates and it was around this time that Corazon saved Law from Marijoies and brought him to that city Robin was teleported too. Corazon kept Law safely hidden. Soone enough. Dragon brought forth a storm and liberated that city(Law was amazed for life). Law escaped the White City and came to Dressrosa, only to find his beloved Corazon murdered by Doffy(persumably through a stab to the heart). That began Law's obsession with hearts, seek out Doflamingo with those grenades and seek the Opi Opi no mi.