Plot hole: Kate is in the future and has named the dog Jack and informs Alex of this in a letter, which he receives in a magic time-travel mailbox. Alex adopts this same dog in the past, before the dog has ever met Kate. Jack tests out the name "Jack" on the dog, and the dog reacts to this name. The dog should not know this name as it hasn't yet met Kate. This is backwards. Alex should have been the one to tell Kate of a name he gave to the dog, and she should have been the one to test it on the dog.
Add timeUncel Mark
Plot hole: When Kate went to the aid of the man lying in the street, she was very upset about his death. Would she not have got his name being that she works in a hospital? Also she could have easily Googled him in 2006 to find out where he was or that he had died already. It seemed an obvious ending that she would save his life to change history.