Cartesian natural transformations

Definition 23.7.4 (Cartesian natural transformation). Given two functors \(F,G\colon I \to T\), a natural transformation \(\alpha \colon F \to G\) is called cartesian if for every morphism \(i \to j\) in \(I\) the naturality square \begin {equation*}

\end {equation*} is a pullback square. We denote by \(\Fun ^{\cart }(I,T) \subseteq \Fun (I,T)\) the wide subcategory spanned by the cartesian natural transformations.