CHICAGO — When it comes to demanding, pushing and sparring, few people could even begin to compare with Rahm Emanuel, the famously foul-mouthed mayor whose no-holds-barred tactics once included sending a dead fish to someone whose work he found lacking.

Then came Karen Lewis.

In Ms. Lewis, the president of the Chicago Teachers Union, which on Tuesday completed a second day on strike from public schools across this city, Mr. Emanuel may have met his match.

She is biting, pushy, witty, unwavering. He is biting, pushy, witty, unwavering. Like him, she appears to hold almost nothing back — a quality that in recent days has infuriated City Hall and thrilled tens of thousands of teachers. She has called Mr. Emanuel a “bully” and a “liar,” someone whose “billionaire friends” are driving his educational philosophy. And that was only last week.