= Solution
Assume $F$ preserves <finite limits>. Given any finite diagram in the <comma category> $(A\downarrow F)$, first take its underlying limit $c$ in $\mathcal C$. Its structure maps $A\to Fc_j$ form a cone, and preservation gives a unique map $A\to Fc$. This object and the underlying projections satisfy the limiting <universal property> in the comma category. The construction also includes the empty diagram, since $F$ sends a <terminal object> to a singleton. Thus $(A\downarrow F)$ has finite limits for every set $A$, proving (ii).
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