Fiberwise construction of limits and colimits of categories

Construction 23.5.1. Let \(p\colon E \to I\) be a cocartesian fibration.

• We write \(E[\cc ^{-1}]\) for the localization of \(E\) at the collection of \(p\)-cocartesian morphisms.
• We write \(\Gamma _I(E)\) for the \(\infty \)-category of sections of \(p\), defined via the following pullback square:

  

• We write \(\Gamma _I^{\cc }(E)\) for the full subcategory of \(\Gamma _I(E)\) spanned by the cocartesian sections: those sections

  

  that are cocartesian functors over \(I\) (i.e., that send arbitary morphisms in \(I\) to \(p\)-cocartesian morphisms in \(E\)).