Can a website find out about a spoofed SSL certificate?

Question

Digital certificates

I'm not very good at cryptography ... I tried to figure out the TLS protocol and all these certificates, but I didn't understand anything right off the bat.
The question is this:
Using some library, I am doing mitm-proxy on my localhost to analyze and replace https traffic. This library automatically generates a certificate, then I import this certificate into the browser so that it does not give a warning, and everything works. But I'm wondering if (purely in theory) the site I go to can find out that I'm using a fake certificate? And if so, is it possible to fix it? For example, the library allows you to specify any certificate for it, and I can export the real certificate from the browser and give it to the library. Or am I misunderstanding something about this protocol?