Isomorphism of categories

An isomorphism of categories $𝖢, 𝖣$ is a functor $𝐹 : 𝖢 \to 𝖣$ with a proper inverse $𝐹 - 1 : 𝖣 \to 𝖢$ such that $𝐹 - 1 𝐹 = 1 𝖢$ and $𝐹 𝐹 - 1 = 1 𝖣$ . cat The categories $𝖢$ and $𝖣$ are thence said to be isomorphic, denoted $𝖢 ≅ 𝖣$ . This is usually too strong, and more typically one deals with the weaker Equivalence of categories $𝖢 ≃ 𝖣$ .