Conditions for certain inclusion functor to preserve internal homs

Question

Suppose that $\mathcal{C}$ is a locally cartesian closed right proper Quillen model category for which all objects are fibrant. Let $x$ be an object of $\mathcal{C}$. Let $\mathbb{F}$ denote the class of fibrations and let $\mathbb{F}_x$ denote the category of all fibrations with codomain $x$. Consider the inclusion functor $$ i : \mathbb{F}_x \hookrightarrow \mathcal{C}/x. $$

Is it true that $i$ preserves all exponentials (i.e., internal homs)?

My question arises from page 54 of the paper https://arxiv.org/pdf/1406.3219.pdf.

Here, the author gives examples of categories $\mathcal{C}$ satisfying the conditions of Theorem 32 (page 53). The relevant condition for us is that $\mathcal{C}$ have a closed class $\mathbb{D}$ of morphisms (Definition 27, page 40). In turn, the relevant condition of this definition is that the inclusion functor $ \mathbb{D}_x \hookrightarrow \mathcal{C}/x$ as above preserve all exponentials for any object $x$.

The author gives the following example:

the category of fibrant objects in any locally cartesian closed model category that is right proper, and in which the cofibrations are the monos.

So, if we take $\mathbb{D}$ to be the class of fibrations, we arrive at my original question.

Answer 1

You also need to assume that cofibrations are monos since this implies that (trivial) cofibrations are stable under pullbacks along fibrations. In turn, this implies that the exponent of two fibrations over $x$ is also a fibration. It is easy to see that the exponent also has its universal property in $\mathbb{F}_x$. It follows that $\mathbb{F}_x$ is closed under exponents in $\mathcal{C}/x$.