Spider-Man doesn't shoot webs from his wrists. That's an inaccuracy propagated by the movies. In the comic books, he wears these devices on his wrists that shoot the web liquid, which Spider-Man makes by... well it was never really explained HOW he made it, but it is made artificially.
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"Meet the new boss, same as the old boss"
"It's all right Ma, I'm only sighing"