Pointwise limits and colimits in categories of sections

We first prove (1). Consider the full subcategory of \(\Cat_{/I}\) spanned by those functors \(u\colon J\to I\) for which the category of sections of \(u^*p\) admits \(K\)-indexed colimits computed pointwise. This subcategory is closed under colimits. Indeed, if \(J\simeq\colim_aJ_a\) in \(\Cat_{/I}\), then The restriction functors in this limit preserve pointwise colimits, so if all the categories on the right admit \(K\)-indexed colimits computed pointwise, then so does the category on the left.Every category is a colimit of simplices, and each \([n]\) is an iterated pushout of copies of \([1]\) along \([0]\). It therefore suffices to consider maps \([0]\to I\) and \([1]\to I\). The first case is precisely the assumption that each fiber admits \(K\)-indexed colimits. A map \([1]\to I\) classifies a morphism \(\alpha\colon i\to j\). The category of sections of the resulting cocartesian fibration sits in a pullback square

The categories in the other three corners admit \(K\)-indexed colimits, the source evaluation \(s\) preserves them, and \(\alpha_!\) preserves them by assumption. Hence the pullback admits \(K\)-indexed colimits computed by the two projections. This proves (1).Statement (4) is [Lurie 2009, Proposition 5.1.2.2], applied to the cartesian fibration \(q\). Passing to opposite categories and using \(K\catop\) as the indexing category gives (2): the functor \(p\catop\colon (\int_I^{\cc}F)\catop\to I\catop\) is a cartesian fibration with fibers \(F(i)\catop\), and Finally, applying (1), with indexing category \(K\catop\), to the cocartesian fibration \(q\catop\colon (\int_I^{\ct}G)\catop\to I\catop\) proves (3), because the opposite of a cartesian transport functor \(\alpha^*\colon G(j)\to G(i)\) preserves \(K\catop\)-indexed colimits precisely when \(\alpha^*\) preserves \(K\)-indexed limits. The pointwise descriptions in these arguments also show that all evaluation functors preserve the indicated limits or colimits.