Initial and terminal objects

Definition 1.7.7 (Initial/terminal objects). An object \(x\) of an \(\infty \)-category \(C\) is called terminal if for every other object \(y\) the hom anima \(\Hom _C(y,x)\) is contractible. We say that \(x\) is initial if for every other \(y\) the hom anima \(\Hom _C(x,y)\) is contractible.