In section 19.2.6 of Lurie's "Spectral Algebraic Geometry," he constructs the smooth projective space, which represents the derived version of [the dual of] the usual functor of points interpretation of projective space. The functorial construction is as follows for convenience:

Construction 19.2.6.1: Fix a nonnegative integer $n\geq 0.$ For every connective $\mathbb{E}_\infty$-ring $A,$ let $X(A)$ denote the subcategory of $(\mathsf{Mod}_A)_{/A^{n+1}}$ whose morphisms are equivalences and whose objects are maps $f : L\to A^{n+1}$ with the following property:

1. The map $f$ admits a left homotopy inverse (that is, exhibits $L$ as a direct summand of $A^{n+1}$).
2. The $A$-module $L$ is projective of rank $1.$

We will regard the construction $A\mapsto X(A)$ as a functor $X : \mathsf{CAlg}^{\textrm{cn}}\to\mathcal{S}.$

My question is about the last sentence: regarding this construction as a functor. Only assigning an $\infty$-groupoid is not enough to describe a functor, and as I understand, even if we describe how $X$ acts on morphisms of $\mathbb{E}_\infty$-rings this is not enough to construct $X$ as a functor.

This is probably standard, but how do we actually confirm that $X$ is a functor of $\infty$-categories? Specifically, I want to understand what needs to be specified in order to define $X$ as a functor, as well as how those details can be verified in this instance.