The man died after being shot by police during a "pre-planned policing operation" near the M62 motorway in Huddersfield.
He has been named locally as Mohammed Yassar Yaqub, aged 27..
Mr Yaqub’s family said they did not want to speak at his home in a cul-de-sac in the Crosland Moor area of the town.
But flowers were taken to the house and a number of relatives arrived, some embracing as they went into the property.
At the scene, a silver Audi with bullet holes in its windscreen was surrounded by other vehicles at the bottom of the off-slip of the westbound carriageway of the M62 on Tuesday morning as police and forensic teams continued their investigation.
West Yorkshire Police said the operation was not terror-related and that a car had also been stopped in the Chain Bar area of Bradford at around the same time.
Five people were arrested across two vehicle stops - three in Huddersfield and two in Bradford.
Several of those arrested needed hospital treatment for injuries “not related to firearm discharge” but no police officers were hurt, a force spokesman said.
People living near the scene of the shooting, which happened at junction 24 at around 6pm on Monday, described the frightening sight of officers “swarming” around a group of vehicles after the incident.
Shahnaz Malik’s house is the closest to the scene and overlooks the slip road from the village of Ainley Top.
The 54-year-old said she was not aware straight away that there had been a shooting, first thinking it was a major accident.
She said: “We do have accidents and we hear sirens. It just seems like there were too many last night.”
Ms Malik said that once she realised something out of the ordinary was under way, she looked over her back fence and saw large numbers of police vehicles and officers.
“There was like a swarm,” she said. “There was a lot. It was dark as well.”
She said: “It was surreal. It was like time had stood still, so many police buzzing around.
“It all happened so quickly. I was just stood there freezing cold.”
Asked why she had looked out in the first place, she said: “It just didn’t feel right. I thought it must be a major accident.
“I didn’t know what I was looking at, to be honest. It was like time had stood still but, obviously, all this movement was going on. I didn’t even occur to me there might have been a shooting.”
An IPCC spokesman said: “An IPCC independent investigation is under way following the death of a man during a pre -planned police incident in Huddersfield. The IPCC understands that the man was shot by a West Yorkshire Police firearms officer shortly after 6pm on Monday 2 January close to Junction 24 of the M62 motorway.
“IPCC investigators remain at the scene and are attending post-incident procedures in order to gather evidence during the early stages of this investigation. The man’s family has been informed.”