---- > [!definition] Definition. ([[category isomorphism]]) > Two [[category|categories]] $\mathsf{C}$ and $\mathsf{D}$ are called **isomorphic** if there exist functors $\mathscr{F}: \mathsf{C} \to \mathsf{D}$ and $\mathscr{G}:\mathsf{D} \to \mathsf{C}$ which are mutual inverses, i.e., $\mathscr{F} \mathscr{G}=1_{\mathsf{D}}$ (the [[identity functor]] on $\mathsf{D}$) and $\mathscr{G}\mathscr{F}=1_{\mathsf{C}}$ (the [[identity functor]] on $\mathsf{C}$). ^definition > [!note] Remark. > This means that both the morphisms and objects of $\mathsf{C}$ and $\mathsf{D}$ stand in one-to-one correspondence, so that two isomorphic categories share all properties defined solely in terms of category theory. This turns out to be a very strong condition that is rarely interesting and rarely satisfied. A more important notion is that of [[category equivalence]]. ^note ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```