Written on the heart

I believe this terminology signifies that which holds a special place for an individual.

People being written on the heart would signify holding them special or dear,  or to be devoted.

The law written on heart in the new covenant may speak of a devotion to it not simply a knowledge of it.

Paul writes about those who “show the work of the law written on their hearts.  I believe this is different than what the new covenant speaks about.  This speaks about those who did not or do not have the law.   

The purpose of law is order and justice.  It's a code of conduct.  Written on heart is a special regard for order, justice, or whatever the “work” of the law is.  This text speaks of the “work” of the law being written, which is probably different than the law being written on the heart—a new covenant provision for Israel, not  a natural condition, true for all nations.  

It is often claimed that by nature the law is written on the heart, but I question that idea.  The "work" of the law is written, not the law itself.  I think of the aim of law is order and justice.

There is a universal recognition that some things are right and some wrong, and instinctively they try to determine who is in the right and who is wrong in a quarrel.  I person will often seek to justify their actions, that they were in the right.

  
The big 3 wrongs are murder, theft, and lying.  The conscience is probably distinct from the “work” of the law; the conscience is that faculty that weighs/ judges actions.