# Equivalence of categories

Two categories $\mathfrak{K}$ and $\mathfrak{L}$ are called equivalent if there are one-place covariant functors $F : \mathfrak{K} \rightarrow \mathfrak{L}$ and $G : \mathfrak{L} \rightarrow \mathfrak{K}$ such that the product $FG$ is naturally equivalent to the identity functor $\mathrm{Id}_{\mathfrak{L}}$ and the product $GF$ to the functor $\mathrm{Id}_{\mathfrak{K}}$; in other words, the categories $\mathfrak{K}$ and $\mathfrak{L}$ are equivalent if there are functors $F$ and $G$ "almost" inverse to one another. Two categories are equivalent if and only if their skeletons are isomorphic.