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According to Wikipedia:

Formally, given two categories C and D, an equivalence of categories consists of a functor F : C → D, a functor G : D → C, and two natural isomorphisms ε: FG→ID and η : IC→GF. Here FG: D→D and GF: C→C, denote the respective compositions of F and G, and IC: C→C and ID: D→D denote the identity functors on C and D, assigning each object and morphism to itself.

Let the notation $C \sim D$ imply that functors $F$ and $G$ exist as above (with natural isomorphisms $\varepsilon$ and $\eta$).

Question: Does $C \sim D$ imply that $D \sim C$?

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Yes, just switch the roles of $F$ and $G$ and invert the natural isomorphisms.

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    As a related question: In Wikipedia's definition of equivalence, could it just have easily been stated that $\varepsilon: ID \rightarrow FG$ and $\eta : IC \rightarrow GF$? That is, instead of $\eta$ having an identity morphism in the domain and $\varepsilon$ having one in the codomain, could they both have had an identity morphism in the domain and the definition for equivalence remain the same? – 2017-02-20
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    Yep, since natural isomorphisms are invertible, it makes no difference. The reason for the convention is that that's how the natural transformations come in an adjunction, where generally they aren't invertible. And every equivalence can be modified into an adjoint equivalence, so there's a close connection-equivalences are roughly just the adjunctions with unit and counit natural isomorphisms. – 2017-02-20
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    Makes sense -- I haven't gotten to adjunctions yet in my textbook so this convention was puzzling. – 2017-02-20