# $\mathbf{Cat}$ the category of the categories is a category

I'm studying this book and I'm trying to prove this assertion the author made:

The identity functor is the identity in this category, i.e., for each category $C$, $Id_C:C\to C$ is the identity functor.

I'm having troubles to prove the associativity of the composition of functors.

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What is your trouble? As it says, functor is just a mapping of objects to objects and arrows to arrows, and they compose by ordinary composition of functions. Therefore, composition of functors inherits associativity because function composition is associative. – Henning Makholm Apr 7 '14 at 18:34
As a side not unrelated to the associativity of functors, I'm not sure I agree with this. I'm fairly certain a "category of all categories" will fall victim to a modified Russell's paradox... one can make it out safely if they work with the category of small categories, say. – Mike Miller Apr 7 '14 at 18:35
Unless you're working in a set theory with a universal set, in which case you just lose the Yoneda lemma or some such :p – Malice Vidrine Apr 7 '14 at 18:45
@HenningMakholm it's true, I didn't notice this fact. – user42912 Apr 7 '14 at 18:54
@HenningMakholm composition of maps is associative and since functors are maps, we're done. – user42912 Apr 7 '14 at 18:55