Hom spaces in (∞, 1)-categories

This is explained in the opening paragraph of HTT.5.5.2; it’s a combination of the fact that both the Yoneda embedding and evaluation in functor categories preserve limits.

This is Cisinski, Corollary 6.3.5. The proof is essentially to show that cocontinuous functors out of presheaf $\infty$-categories admit right adjoints given by the usual formula, so that $Hom_C$ has a left adjoint if $C$ is cocomplete, and then to use the Yoneda embedding for a general $C$.