Activating relay from 2 sources

I currently have an aftermarket remote lock/unlock system for a car. It works by triggering a Bosch relay that triggers the door lock. I have developed an Arduino Uno app that will allow me to trigger the Bosch relay from an iPhone. I found a electrical schematic on the Henry’s Bench website for using the Uno to control the Bosch relay. The schematic below is my proposed interface between the existing remote system and the Uno/iPhone system. Both systems close ground to activate the relay.

I am a true electronics novice but is there going to be a issue with activating the relay (Bosch pin 85) from 2 different sources?

Is there going to be a issue with activating the relay (Bosch pin 85) from 2 different sources?

The relay won't mind, what you can't be sure of is whether the existing control box will object to its output being pulled low by something else. You can guard against this possible problem by putting a diode between the relay coil and the existing control box with the anode on the relay, cathode on the control box.
You should not have the relay current flowing through the Arduno board. I suggest connecting the power to the Arduino from the Bosch controller, not from the battery, and connect the transistor to 0V at the same point or just before where the 0V connects to the Arduino. The power for the whole system should come form the same source.

voltage and current ratings. The common 1N4001 will be fine for almost any relay winding you can think of.

How the need for a second diode?

Not needed.
The problem is you don't know what the output of the Bosch controller is, so the diode is a precaution. I suspect you don't need it at all, but I suggested it just in case. For your added transistor you know what the output is, it will be open collector / open drain, which does not need isolating from the Bosch output.