Python 2, 39 37 bytes

thanks to @Mr.Xcoder for -2 bytes

s=lambda f,k:s(f,f(k))if k-f(k)else k

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Python 2, 39 37 33 bytes

thanks to @Mr.Xcoder for -2 bytes

s=lambda k:s(f(k))if k-f(k)else k

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assumes the black-box function to be named f

Python 2, 39 bytes

s=lambda f,k:k if k==f(k)else s(f,f(k))

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Python 2, 39 37 bytes

thanks to @Mr.Xcoder for -2 bytes

s=lambda f,k:s(f,f(k))if k-f(k)else k

Try it online!