Why having finite products and equalizers implies having terminal object?

Question

So, I am reading Steve Awodey "Category Theory", and what he claims is that if category has finite products and equalizers then it has terminal object. I am not so sure why is this true. I mean, he says "a category", which can be large. So, a terminal object is a product of all objects in a category, so possibly infinite products also have to exist. Why is this true? Thanks!

Answer 1

The terminal object is not the product of all objects, it is the product of no objects. Already having finite products implies having a terminal object, by taking the product of the empty set of objects.

Why is the terminal object the empty product? See this question.

Answer 2

"Finite products" includes products of families of objects of any finite size... including zero.

The terminal object is the product of the empty family; i.e. a family of zero objects.