Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 2 / b / i / Solution

Assume $F$ preserves existing categorical limits. For a diagram in a category $E:I\to \mathcal{B}$ with categorical limit cone ${\ell }_i:L\to E_i$, its image $F{\ell }_i$ is a categorical limit cone of $FE$. For any $C\in \mathcal{C}$, a morphism $C\to FL$ is therefore uniquely equivalent to a compatible family of morphisms $C\to F{E}_i$. In the Category of sets, compatible families are exactly the categorical limit of the resulting hom-sets. ThusThe maps are induced by $F{\ell }_i$, so $\mathcal{C}(C,F-)$ preserves the given limit. Applying this to every existing categorical limit proves the implication, with no completeness hypothesis on $\mathcal{B}$.