Emulating Physical Switch Connecting Two Voltage Sources
so the control wires idles at about 8V and the horn relay has about 12V on it (no suprises there the horn relay resistance is only a few hundered ohms) 

a tranbsistor array like ULN2003 is an interesting choice for the pull-down, but it could be made to work.  darlington transistors have an inherent 0.6V minumum output voltage becasue the output transistor gets most of its base current from the collector terminal, you'll probably need to compensate for this by reducing the resistance of the resistors.

for the pull up to +12 I'd use a PNP transistor.







simulate this circuit &ndash; Schematic created using CircuitLab

Sorry about the cramped schematic, circuit lab would not allow me to draw "wire"

the diode is needed to prevent Q1 from self-distructing when someone sounds the horn. Q2 could be replaced with one of the channels of the transistor array.
if you are not installin in the steering wheel you coul connect to the other side of the horn relay (or a simialr 12V source) and not need the diode.