When Burnham is speaking with Amanda, Amanda lowers and folds her umbrella. The umbrella subsequently disappears without a trace.
When Sarek gives Burnham her Starfleet badge, it is a gold command-division badge despite her uniform being science-division silver. Division badges and uniform colors are supposed to match. In the next scene, she has changed her uniform to command-gold and maintains the gold badge. The very next scene, she is once again in her silver science-division uniform with a proper matching silver badge.
Portions of Tilly's hair had been cut short to pass as her Terran counterpart. Her hair is suddenly back to its long, uniform length.
At the end of the episode, when they drop out of warp, Saru's right arm is raised but resting on the chair when the camera angle changes.
Early in the episode, Georgiou said she was giving the Federation an advantage in the war. She says "us," but her mouth says "them."
At the end, Saru asks if they're out of the Sol System, to which Detmer answers in the affirmative. However, on the screen Neptune and several moons can be clearly seen. Astronomers place the edge of the Sol System at around 97 AU (according to NASA) while Neptune is only about 30 AU from the Sun. They weren't out of the Sol System by a long shot and hadn't even made it to the orbit of Pluto or the Oort Cloud.
Burnham questions the meaning of the phrase, "Isik for your thoughts," which was identified in Despite Yourself (2018) as "an old Vulcan saying." Her foster mother, Amanda, tells her she doesn't know what an Isik is, it was just something her mother used to say. Amanda is human, not Vulcan.
Tilly is made a full officer and put into the command training program, yet she wears a bronze engineering uniform and badge rather than a gold command uniform and badge.