Does derived hom commute with homotopy limits?

Question

Suppose that $\mathcal{V}$ is a symmetric monoidal model category, and that $\mathcal{C}$ is a $\mathcal{V}$-enriched model category. Write $\Bbb{R}\!\operatorname{Hom}(-,-)$ for the derived Hom functor $$ \Bbb{R}\!\operatorname{Hom}(-,-) : \operatorname{Ho}(\mathcal{C})^{\textrm{op}}\times\operatorname{Ho}(\mathcal{C})\to\mathcal{V}. $$ Suppose also that $\mathcal{C}$ (respectively $\mathcal{V}$) has functorial cofibrant and fibrant replacement functors, denoted $Q_\mathcal{C}$ and $R_\mathcal{C}$ (respectively $Q_\mathcal{V}$ and $R_\mathcal{V}$), so that $\Bbb{R}\!\operatorname{Hom}(X,Y)$ may be computed as $\mathcal{C}(Q_\mathcal{C}(X),R_\mathcal{C}(Y)).$

Let $X\in\mathcal{C},$ let $F : D\to\mathcal{C}$ be a diagram in $\mathcal{C},$ and suppose that $\operatorname{holim}F$ and $\operatorname{holim}\Bbb{R}\!\operatorname{Hom}(X,-)\circ F$ exist.

Question: Is it true that $$ \operatorname{holim}\left(\Bbb{R}\!\operatorname{Hom}(X,-)\circ F\right)\simeq\Bbb{R}\!\operatorname{Hom}(X,\operatorname{holim}F)? $$ If this is true, what is the proof (preferably a proof using the language of model categories, as opposed to a proof using $\infty$-categories). If this is not true in general, I would be interested in knowing what general hypotheses could be placed on the objects/functors/categories involved which would guarantee that there is such an equivalence.

I've heard that in the language of $\infty$-categories, we do have commutativity of limits and homs, which makes me suspect that this statement at the level of model categories should hold. However, I'm not sure if the translation is so direct, and it makes me a bit suspicious that I have not been able to find a statement like the above anywhere in my searches of the literature, and my attempts to prove the statement have not been fruitful.

Thanks!

Answer 1

Yes, this is always true.

Replacing $X$ by its cofibrant replacement if necessary, we can assume $X$ to be cofibrant. In this case, $\def\Hom{\mathop{\rm Hom}} \Hom(X,-)\colon C→V$ is a right Quillen functor. The right derived functor of this right Quillen functor computes the derived hom $\def\RHom{\mathop{\rm RHom}} \RHom(X,-)$.

We can now conclude by invoking the fact that right derived functors of right Quillen functors preserve homotopy limits.

The latter fact can be proved as follows. For simplicity of exposition, suppose that the injective model structure on D-indexed diagrams valued in $C$ and $V$ exists. (This is true in almost all practical cases, and the assumption can be eliminated anyway by working with more complicated models for homotopy limits.)

Replacing $F$ by its fibrant replacement if necessary, we can assume $F$ to be injectively fibrant. Then $\RHom(X,-)∘F$ is also injectively fibrant.

Homotopy limits of injectively fibrant diagrams can be computed as ordinary limits. Thus, the statement becomes $$\lim(\Hom(X,-)∘F) ≅ \Hom(X,\lim F),$$ which is a true 1-categorical fact.