I have two questions about images of morphisms of quasi-categories.

Suppose that $f\colon X \to Y$ is a morphism of quasi-categories.

- Suppose that we calculate the image of $f$ in the category $\mathsf{QCat}$, using the universal property of image. Will this be a Joyal-fibrant replacement of the image as calculated in $\mathsf{SSet}$?
- If $X = N(C)$ and $Y = N(D)$ are (nerves of) categories, will the image of $f$ be the nerve of the essential image of the functor $C \to D$?

In case the answer to one of these questions is no, can the statements be adjusted to become true?