[[Category theory MOC]]
# Equivalence of categories

**Equivalence** of categories is a weakening of [[isomorphism of categories]] which in the majority of cases is actually more appropriate.
Equivalent categories are “almost” the same in that their categorical properties coïncide.
An **equivalence** of categories $\cat C,\cat D$ is a pair $F : \cat C \leftrightarrows \cat D : G$ of functors such that
$$
\begin{align*}
GF &\simeq 1_{\cat C}, & FG \simeq 1_{\cat D}
\end{align*}
$$
where $(\simeq)$ denotes [[natural isomorphism]].
This is reminiscent of [[Homotopy equivalence]].
We also see equivalence of categories is a special case of an[[Adjoint functor|adjunction of functors]] for which the unit and coünit are isomorphisms. 


## Results

- [[Categories are equivalent iff they have isomorphic skeleta]]

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