When the T-1000 goes to Todd and Janelle Voight's home, why does he ask to keep the photo of John?
Presumably he can remember the photo in perfect detail?
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The T-1000 is using the photo as a tool to help him search for John, showing it to passers-by in the hopes that one of them will recognise him. This works remarkably well.
EXT. 7-ELEVEN STORE - DAY
Officer X has stopped two young girls in front of a 7-Eleven. He is leaning out the cruiser window and showing them the picture of John.
The first girl nods.
FIRST GIRL: Yeah, he was here about fifteen minutes ago. I think he said he was going to the Galleria.
and from the film's official novelisation
He [the T-1000] approached a few kids near Perry’s Pizza on the second floor, then showed them the photograph. They could only shrug. He glanced over at a security guard who was eyeing him curiously. Austin considered the possible alternatives and a second later walked up to the man and showed him the photo, asking him to look out for the boy.