If your category has pullbacks and equalizers, do you get products?

Question

I've proven that products + pullback gives equalizers, and products + equalizer gives pullback. So, can we get products out of pullback + equalizer?

Answer 1

Short answer: no. There is actually a very trivial counterexample. Take the category $\mathbf 2$ with just two objects (say $A$ and $B$) and only the identity arrows. This category has all pullbacks and equalizers, but there is no product $A \times B$. This also indicates the general problem: a product is a limit of a disconnected diagram, while equalizers and pullbacks are limits of connected diagrams.

You may be interested in the following though. The following are equivalent for any category $\mathcal{C}$:

1. $\mathcal{C}$ has all finite limits;
2. $\mathcal{C}$ has finite products (including the empty product, i.e. terminal object) and equalizers;
3. $\mathcal{C}$ has pullbacks and a terminal object.

See for example, this nLab page. So this means that we can either use finite products + equalizers to build any finite limit, or we can use pullbacks + a terminal object.

Exercise: suppose we have a category $\mathcal{C}$ that has pullbacks and a terminal object, how can you use this to form the product of two objects $A$ and $B$?