In chapter I, section 9, proposition 5 in Mac Lane and Moerdijk's Sheaves in Geometry and Logic, it is stated that if $f : B' \to B$ is a morphism in a complete category $\mathcal{C}$ and the category $\mathcal{C}$ and the slice categories $\mathcal{C}/B$ and $\mathcal{C}/B'$ are (all three) cartesian closed, then pullback along $f$ preserves all colimits which exist in $\mathcal{C}/B$.
- Why should $\mathcal{C}$ be complete? It must have pullbacks (along $f$), but is this completeness hypothesis needed anywhere?
- This proposition holds because if $\mathcal{C}/B$ is cartesian closed then pullback along $f$ has a right adjoint and then preserves colimits (this is a theorem appearing right before this proposition), so why are also $\mathcal{C}$ and $\mathcal{C}/B'$ assumed to be cartesian closed?
Thank you.