Equivalence of categories

In category theory, equivalence is a weaker and more prevalent notion of "sameness" than isomorphism. For two categories to be isomorphic, they must have two functors $F:𝐂\to 𝐃$ and $G:𝐃\to 𝐂$ such that their composition is the identity functor; the functors between a pair of equivalent categories need only be naturally isomorphic to the identity functor.

Unsurprisingly, equivalence of categories defines an equivalence relation.[cite:@riehl2017category]

Definition

An equivalence of categories, denoted $𝐂\simeq 𝐃$, consists of functors $F:𝐂\to 𝐃$ and $G:𝐃\to 𝐂$ equipped with natural isomorphisms $\eta :\text{Id}𝐂\cong FG$ and $\epsilon :GF\cong \text{Id}𝐃$.

It should be emphasised once more that both compositions of $F$ and $G$ are naturally isomorphic to the identity functor, not equal. Just like isomorphism of objects, this is considered "close enough" for the majority of practical purposes.

Remarks