Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 26 / 3 / i / Solution

Assume $F$ preserves finite limits. Fix any set $A$ and a finite diagram of objects $(C_j,u_j)$ in $(A\downarrow F)$. Let $L$ be the categorical limit of its underlying diagram in $\mathcal{C}$, with projections $p_j$. Preservation gives $FL\cong {\lim }_{j}F{C}_j$. The maps $u_j:A\to F{C}_j$ form a compatible cone, so determine a unique $u:A\to FL$ with $F{p}_j\circ u=u_j$.

The object $(L,u)$ is the categorical limit in the comma category. Indeed a cone from $(B,v)$ gives a unique underlying $h:B\to L$, and the identities $F{p}_{j}Fhv=F{p}_{j}u$ for all $j$ force $Fhv=u$ by the categorical limit's uniqueness. For the empty diagram, $L$ is terminal and $FL$ is a one-element set, giving the unique map from $A$. Thus $(A\downarrow F)$ has all finite limits. This proves $\boxed{\text{(i)}\Rightarrow \text{(ii)}}$.