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The hose that opens up infront of the battery and behind the lamp assembly is the snorkel for the airbox. It pulls air from there through the resonator, which is you have pictured behind the driver's side splash shield / bumper area.
The electrical connection is more curious - I have a 2002 Corolla, and I don't remember anything like that. It looks to be tied to the chassis immediately above. Possible you can take a picture of the bottom of the connector?
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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