essentially small infinity category as locally small infinity category

Let \(\mathcal{C}\) be an essentially small infinity category.
Then \(\mathcal{C}\) is also locally small

by joyal equivalence as fully faithful essentially surjective functor we may assume that \(\mathcal{C}\) is a small infinity category.
Then \(\mathrm{map}_{\mathcal{C}}(x,y)\) is a simplicial subset of \(\mathrm{Fun}(\Delta^1, \mathcal{C})\) which is small by infinity functor category of small categories as small category