Australian Category Seminar

Proposition 1. A category has finite connected limits iff it has pullbacks and equalizers.

Proposition 2. A functor between finitely complete categories preserves finite comnected limits if it preserves pullbacks.

Example. Categories $\mathcal{A}$ and $\mathcal{B}$ have finite connected limits and $F \colon \mathcal{A} \to \mathcal{B}$ preserves pullbacks but doesn't preserve equalizers. (Let $f \colon H \to G$ be a non one-one group homomorphism. Build a category $\mathcal{G}$ out of $G$ by considering it as a one object category and formally adjoining an initial object. Do the same for $H$ and define a functor in the obvious way. This last mentioned functor is an example of the above.)

Question: Why are the finite limits needed in proposition 2? Or rather, what is the $A$ and $B$ have limits of finite connected diagrams which admit a cocone, then $F$ preserves these iff $F$ preserves pullbacks.