Terminator in The Matrix?

I've been thinking about what would happen if the Reprogrammed Terminator (Terminator 2) were in the first Matrix movie, and that Trinity is let's say erased. How different would the movie be then? What would a relationship between the T-800 and Neo be like? I'm writing all of this, as I've seen fan theories that explain that Terminator and The Matrix exist in the same universe.

and I feel like the T-800 and Neo fighting side-by-side would be badass as hell.

What do you guys think?