Of all the fighters in the entire Dragon Ball franchise, few are as powerful as the composite warrior Gogeta. Formed by Goku and Vegeta merging together through the Fusion Dance, the canonical iteration of the character was formally introduced in 2018's Dragon Ball Super: Broly after a non-canonical appearance in 1995's Dragon Ball Z: Fusion Reborn. While Gogeta possesses many of the same incredible techniques and abilities as Goku and Vegeta, one lingering question remains after the character's sole canonical appearance: Can Gogeta achieve the Ultra Instinct combat transformation?
Introduced in the final story arc of the Dragon Ball Super anime, Ultra Instinct has users achieve a sort of divine awareness of the world around them. This gives them a prodigious level of defensive ability as they intuitively sense incoming attacks and react unconsciously. In addition to the defense boost, Ultra Instinct also significantly augments the user's combat strength, with Goku capable of landing attacks that completely knocked around his Tournament of Power opponent Jiren.
Following the anime's conclusion, the Super manga would go on to depict Goku gaining greater mastery over Ultra Instinct. He's now able to achieve new abilities with it and trigger the transformation at will, rather than only when unconsciously pressured to exceed his normal physical limits.
What puts the possibility of an Ultra Instinct Gogeta into question is that Vegeta has never been able to achieve Ultra Instinct for himself. While Vegeta's level of active strategy has made him one of the deadliest fighters in the Dragon Ball Multiverse, it has also prevented him from mentally letting go and allowing the divine combat energy flow through him to trigger the Ultra Instinct transformation. While Vegeta would also grow much stronger in the manga beyond the Tournament of Power, Ultra Instinct continues to elude him, much to his frustration.
With both Vegeta and Goku capable of transforming into Super Saiyan Blue, Gogeta is similarly able to take on the powerful form but has never visibly attempted reaching Ultra Instinct. While Goku's Ultra Instinct ability suggests Gogeta is capable of achieving the form as well, this would require that he reins in the aspect of his consciousness informed by Vegeta. With Gogeta serving as the personification of how powerful Goku and Vegeta could be combining their might to work together, this is possible, but Gogeta may struggle at maintaining the transformation because of Vegeta's significant influence on the composite warrior.
With all of Goku and Vegeta's moves at his disposal at a significantly higher level, it's still seems virtually guaranteed that Gogeta would be able to achieve the Ultra Instinct transformation for himself. While he may be a bit shakier at it due to Vegeta, Gogeta is definitely aware of his own capabilities and weaknesses. He could compensate for these weaknesses to take on the form for himself. Already unstoppable as a Super Saiyan Blue, Gogeta would be a destructive force to be reckoned with at Ultra Instinct.