Unstraightening functors

Notation 23.2.9. We will denote the inverses to the straightening functors \(\Str ^{\cc }\) and \(\Str ^{\ct }\) by \[ \Un ^{\cc }\colon \Fun (C,\Cat _{\infty }) \iso \Cocart (C) \qquadtext { and } \Un ^{\ct }\colon \Fun (C\catop ,\Cat _{\infty }) \iso \Cart (C), \] and refer to these as (cocartesian/cartesian) unstraightening. Given a functor \(F\colon C \to \Cat _{\infty }\), we may describe its unstraightening \(\Un ^{\cc }(F)\) informally as follows:

• An object in \(\Un ^{\cc }(F)\) is a pair \((x,a)\), where \(x\) is an object of \(C\) and \(a\) is an object of \(F(x)\);
• A morphism in \(\Un ^{\cc }(F)\) from \((x,a)\) to \((y,b)\) is a pair \((f,\phi )\), where \(f\colon x \to y\) is a morphism in \(C\) and \(\phi \colon f_!(a) \to b\) is a morphism in \(F(y)\). Here we write \(f_!\) for the functor \(F(f)\colon F(x) \to F(y)\) induced by \(f\) using the functoriality of \(F\).
• Composition of \((f,\phi )\colon (x,a) \to (y,b)\) and \((g,\psi )\colon (y,b) \to (z,c)\) is defined as \((g \circ f, \psi \circ g_!(\phi ))\).

The functor \(\Un ^{\cc }(F) \to C\) is given by \((x,a) \mapsto x\) and \((f,\phi ) \mapsto f\). A dual description holds for \(\Un ^{\ct }(F)\).