Help with identifying hose adventra.

[Moosejaw]

While changing out coolant hoses and heater tap on an alloytec

i noticed a hose at the rear of engine bay that i do not remember disconnecting nor can i see where it should go onto.
It has a bit of a hooked end and does not rise up to far above engine and it runs down firewall into tranny tunnel on passenger side and joins into metal fitting. Have not tracked it further yet.
I am trying to understand what to hook it up to.

 

[Fu Manchu]

Transmission breather hose

 

[Moosejaw]

Transmission breather hose

Thanks for that clarification very appreciated.