Introduction to Category Theory/Equivalence of categories
Given two categories C,D, we can have functors going both ways. If there were two functors f:C ->D and g:D->C such that the two compositions gf: C->C and fg: D->D are naturally isomorphic to the identity functors in C and D respectively, then we say that "C and D are equivalent categories"; and that "f and g give an equivalence of categories C and D".
1. Take your favourite field (for example, the field of real or complex numbers). Consider the category C of finite-dimensional vector spaces over it. Then consider the category D of finite-dimensional vector spaces together with the datum of a fixed basis. Are they equivalent?