= Equalizer preservation by pullback-preserving functors
If $\mathcal C$ has finite <categorical limits>, every <functor> $F:\mathcal C\to\mathcal D$ preserving <pullbacks in a category> also preserves <equalizers>, even when it does not preserve a <terminal object>. Factor it through the <slice category> $\mathcal D/F1$ by sending $A$ to $F(A\to1)$. This lifted functor preserves pullbacks because the slice <forgetful functor> reflects them; it also preserves the terminal object, so it preserves finite limits. The slice forgetful functor preserves equalizers, completing the proof.
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