# Introduction to Category Theory/Equivalence of categories

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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*".

## Exercise

[edit | edit source]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?