The Stacks project

Definition 4.29.4. Two objects $x, y$ of a $2$-category are equivalent if there exist $1$-morphisms $F : x \to y$ and $G : y \to x$ such that $F \circ G$ is $2$-isomorphic to $\text{id}_ y$ and $G \circ F$ is $2$-isomorphic to $\text{id}_ x$.