[[Category theory MOC]]
# Isomorphism of categories

An **isomorphism** of [[Category|categories]] $\cat C, \cat D$ is a [[functor]] $F : \cat C \to \cat D$
with a proper inverse $F^{-1} : \cat D \to \cat C$ such that $F^{-1} F = 1_{\cat C}$ and $FF^{-1} = 1_{\cat D}$. #m/def/cat
The categories $\cat C$ and $\cat D$ are thence said to be **isomorphic**, denoted $\cat C \cong \cat D$.
This is usually too strong, and more typically one deals with the weaker [[Equivalence of categories]] $\cat C \simeq \cat D$.


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#state/tidy | #lang/en | #SemBr