A short proof of the straightening theorem

We provide a short and reasonably self-contained proof of Lurie’s straightening equivalence, relating cartesian fibrations over a given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞</mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with contravariant functors from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞</mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category of small <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞</mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories.