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In ordinary category theory it is a well-known and important fact that the $\hom$ bifunctor into $\text{Set}$ preserves limits.

I am unable to find a reference for the corresponding fact in infinity category theory nor am I able to write down a proof myself.

I am looking for a statement of the following form:

Let $\mathcal{C}, \mathcal{D}$ be $(\infty, 1)$-categories (considered as a weak Kan complex preferably) and let $F: \mathcal{D} \to \mathcal{C}$ be a diagram. Then, for any object $X \in \mathcal{C}$, we have a natural equivalence between $\lim \hom(X, F(D))$ and $\hom (X, \lim F(D))$ where each $\hom$ is considered as a weak Kan complex.

I would presume that HTT would have a statement to this effect, but I am unable to find one.

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  • $\begingroup$ I thought this was the definition of "limit". $\endgroup$ – Theo Johnson-Freyd Jan 8 at 1:01
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    $\begingroup$ @TheoJohnson-Freyd Limits can be defined as terminal cones, in which case something has to be proved about the relationship between $\infty$-categories of cones and the mapping space functor. They can be defined as certain Kan liftings, in which case the connection to hom is still less obvious. I am not sure whether anyone writing foundational material literally defines a limit cone as one sent to a limit in spaces under hom. This would depend on giving an explicit definition of limits in spaces, which is not so easy as in sets. $\endgroup$ – Kevin Arlin Jan 8 at 1:10
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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 $C$.

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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.

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