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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!

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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.

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"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.

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