= 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 $FL\cong\lim_jFC_j$. The maps $u_j:A\to FC_j$ form a compatible cone, so determine a unique $u:A\to FL$ with $Fp_j\circ u=u_j$.
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 $Fp_j Fh v=Fp_j u$ for all $j$ force $Fh v=u$ 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 $\boxed{\text{(i)}\Rightarrow\text{(ii)}}$.
Back to article page
