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Perhaps we know what a diagram(like a commutative diagram) in a category is. I would like to know a definition of it.

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Here wiki got a nice article, hope that will help.

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@MakotoKato Wikipedia says:"The actual objects and morphisms in J are largely irrelevant, only the way in which they are interrelated matters". This simply means that the actual names of objects and morphisms are irrelevant (of course), but the interrelations matter. This is a common theme in CT. I think you confuse the "diagram" which is a functor with its image in the codomain category. It is true that the image counts more, but we still need the functor so that we can compose functors (and -in a sense- transfer the diagrams) and find if and how limits are preserved or created. –  magma Jan 9 '13 at 3:10
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@Makoto I don't agree with any definition that says that a diagram is necessarily a substructure of a category. Like it or not, diagram is now a technical word in category theory, and it is a synonym for functor. –  Zhen Lin Jan 9 '13 at 7:46
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@MakotoKato, You asked for a definition of diagram. You got one, the one used by pretty much everyone. Now, it seems that you do not like this definition... Either provide a definition for what a diagram is for you ands then ask for a definition for that (and, sure, this would be rather stupid...) or... well, I do not see any other option. In any case, this discussion has clearly stopped being useful. Start a chat room and debate there. –  Mariano Suárez-Alvarez Jan 10 '13 at 21:09
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As you know well, this is not a site for discussions. You asked for a definition of diagrams and you got the standard one (which you of course knew well...) But obviously the point of your question was not asking for the definition of diagram but debating it or something. That is off-topic. You are free to engage anyone who wants to be engaged in a discussion about what you feel should be the definition of diagrams elsewhere. –  Mariano Suárez-Alvarez Jan 11 '13 at 7:01
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@MarianoSuárez-Alvarez I don't particularly want to discuss it. Since ZhenLin didn't answer, it's unlikely that there will be further discussion. However, someone may want to post his opinion. I don't think it's a good idea to suppress it. –  Makoto Kato Jan 11 '13 at 8:52
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