The phrase "becoming wise to" is correct and usable in written English.
It can be used when someone is starting to understand or become aware of something, often in a context where they are gaining insight or knowledge about a situation.
Example: "After months of manipulation, she finally started becoming wise to his deceitful tactics."
Alternatives: "becoming aware of" or "gaining insight into".
Women are very clear that they will not settle for a system that does not listen to them or take account of their lives; the legal system is becoming wise to that fact.
While most fans might not yet be able to tell their J-League from their K-League Classic, the local game is becoming wise to the idiosyncrasies and opportunities of Asia.
The evidence is that the mainstream supermarkets are becoming wise to the negative effect on "trust" caused by constantly shouting at their customers through the price promotions megaphone.
It seems the public is becoming wise to their arguments.
After becoming wise to Jenna's plan to crash the event with another surprise wedding attempt, Liz conspired with the lights person to make the lighting like a grocery store milk aisle, which is so universally unflattering that Jenna would never be willing to go up on stage.
I realized that the way to do this film was to use the old colonialist story as a personal journey that goes from being naïve to becoming wise.
Jazz was becoming wise, strange and bitter, a dream world full of wide-open space or density, meta-thoughts and paradoxical impulses.
What he is presenting, then, is not a method of discovery but a process of becoming wise.
In the current study, those who more strongly defined aging in terms of becoming wiser, more appreciative, and developed also perceived to be less in control over negative experiences related to aging and generated more negative emotional responses and attributed more negative consequences to aging.
"It seems that people are actually becoming wiser as they age," he said.
Clubs are becoming wiser regarding foreign player acquisition, operating their own trusted scouts and advisers rather than being at the mercy of agents who were able to extract overpriced deals for their clients.
