Category has Products and Equalizers iff Pullbacks and Terminal Object

Theorem

Let $\mathbf C$ be a metacategory.

Then the following are equivalent:

$(1): \mathbf C$ has all finite products and equalizers.

$(2): \mathbf C$ has all pullbacks and a terminal object.

Proof

$(1)$ implies $(2)$

Suppose $\mathbf C$ has all finite products and equalizers.

That $\mathbf C$ has pullbacks follows from Pullback as Equalizer.

That $\mathbf C$ has a terminal object follows by Empty Product is Terminal Object.

$\Box$

$(2)$ implies $(1)$

Suppose $\mathbf C$ has all pullbacks and a terminal object.

From Category has Finite Products iff Terminal Object and Binary Products, it suffices to check that $\mathbf C$ has binary products.

That this is the case follows from Product as Pullback.

That $\mathbf C$ has equalizers follows from Equalizer as Pullback, as it is established that $\mathbf C$ has finite products.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (next): $\S 5.4$: Proposition $5.14$