= Solution
Let $!_A:A\to1$ be the unique arrow to the <terminal object> of $\mathcal C$. Define
$$
\widehat F(A)=(FA\xrightarrow{F!_A}F1),\qquad \widehat F(f)=Ff.
$$
This is a <functor> into the <slice category> $\mathcal D/F1$, because $!_Bf=!_A$, and $U_{F1}\widehat F=F$. The <terminal object> of this slice is $1_{F1}:F1\to F1$, exactly $\widehat F(1)$, so $\widehat F$ preserves the <terminal object>.
The composite $U_{F1}\widehat F$ preserves <pullbacks> by hypothesis, and $U_{F1}$ reflects <pullbacks> by part (b). Part (a), restricted to the <pullback> shape, therefore shows that $\widehat F$ preserves <pullbacks>. The allowed finite-limit criterion now gives that $\widehat F$ preserves all finite <categorical limits>, and hence <equalisers>. Since $U_{F1}$ preserves <equalisers>, their composite does too. Thus
$$
\boxed{F\text{ preserves equalisers.}}
$$
This is <equalizer preservation by pullback-preserving functors>. The factorization supplies terminal-object preservation only in the slice; none is asserted for the original $F$.
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