Exchange 2013 (internal) to Exchange 2010 (internal) Connector

I've found bits and pieces on how to do this, but still cannot get it to work. Here's the scenario:
We've split Company A into Company A and Company B.  Company A (Exchange 2010) is configured and is working properly.  Company B (Exchange 2013) is configured and working properly (internal mail, won't send external because of connector issue).  What I want to do is route Company B's mail through Company A's Exchange.  The reason being because we already have our mxDNS records for Company B pointed to Company A, and I also wouldn't need to open another port up on the firewall or use another public IP up.   So what I'm trying to do is make a connector between the two Exchange Servers (separate domains, same network\dhcp) so we only have one public facing email server.  Here's what I've tried so far on Company B (Exchange 2013):  Relay 1 - Type: Custom, Delivery: MX record associated with recipient domain, Address Space: IP of Company A Exchange 2010, Source Server: Company B's Exchange 2013 server.  THIS DIDN'T WORK AT ALL.   Relay 2 - Type: Internet,    Delivery: MX record associated with recipient domain, Address Space: * (all), Source Server: Company B's Exchange 2013 Server.  THIS ALLOWED INTERNAL MAIL BUT NOT EXTERNAL.  So in a nutshell my questions are:  Whats wrong with my connectors on Exchange 2013?  Do I need to make a receive and\or send connector on Company A's Exchange 2010 server to pass Company B's Exchange 2013 mail?  Is what I want to do possible?  I appreciate any help and I will continue to Google but like I said, I'm only getting little bits here and there and its not working when I put it all together.  Thank You-  Daniel