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I am reading a nice book called category theory by S. Awodey and on page 104 he proves the above statement, which unfortunately for me is a bit sterile. I was wishing to see a picture-like construction of limit out of products and equalizers, preferably demonstrated for a little drawable diagram. I would be very thankful if some one could help me with that. Thanks.

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What does products and equalizers of size $k$ precisely mean? – Stefan Hamcke Aug 13 '13 at 15:31
If k is an (infinte) cardinal, limits of size k are limits of diagrams whose index category has at most k morphisms (or objects). – Martin Brandenburg Aug 13 '13 at 15:44

This is explained in detail in the book by Mac Lane, Categories for the working mathematician, Theorem V.2.1.

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