Functoriality of Hom

Question

If $C$ is a category then $\mathrm{Hom}:C^\mathrm{op}\times C\to \mathrm{Set}$ is covariant. What happens when $C$ is a higher category? For example, should $-^\mathrm{op}$ be replaced by $-^\mathrm{op(1)}$ or something else in order for $\mathrm{Hom}$ to remain covariant?