While $(\infty, 1)$-categories continue to scare me (but also bring me joy!), it is almost frightening how naturally $(\infty, 2)$-categories seem to pop up if you're interested in $(\infty, 1)$-categories. Here is one such situation where they seem to be tacitly appearing in Haugseng's An allegedly somewhat friendly introduction to $\infty$-operads, p. 33.
There is a theorem by Lurie (Theorem 3.4.4) stating the following:
Theorem. The $\infty$-category of $\infty$-operads $\mathsf{Opd}_{\infty}$ admits a symmetric monoidal structure with the Boardman-Vogt tensor product which preserves colimits in each variable.
The immediate corollary (Corollary 3.4.5) is:
Corollary. The $\infty$-category $\mathsf{Opd}_{\infty}$ admits internal Homs $\mathsf{ALG}_{\mathscr{O}}(\mathscr{P})$ with natural equivalences $$\mathsf{Alg}_{\mathscr{O} \otimes \mathscr{P}}(\mathscr{Q}) \simeq \mathsf{Alg}_{\mathscr{O}}(\mathsf{ALG}_{\mathscr{P}}(\mathscr{Q})).$$
From this Haugseng then deduces that the underlying $\infty$-category of $\mathsf{ALG}_{\mathscr{O}}(\mathscr{P})$ is $\mathsf{Alg}_{\mathscr{O}}(\mathscr{P})$.
I'm a little confused about how the corollary arises. First, I don't know if $\mathsf{Opd}_{\infty}$ is presentable (is it?) - but if it is, then the theorem is essentially saying that $\mathsf{Opd}_{\infty}$ is presentably symmetric monoidal and we can take $\mathsf{ALG}_{\mathscr{O}}(-)$ as a right adjoint of $- \otimes \mathscr{O}$ by the Adjoint Functor Theorem. But from this I don't know how to deduce the equivalence. We work inside $\mathsf{Fun}^{\mathsf{Opd}_{\infty}}(-,-)$ if we work with $\mathsf{Alg}$ but an $(\infty, 1)$-adjunction only makes statements about the $\mathsf{Hom}$ anima. So it feels like one needs a $(\infty, 2)$-adjunction to make statements about $\infty$-functor category functors.
Are there some references or explanations to clear up these confusions of mine? While this fact is mentioned in the introduction of Lurie's Higher Algebra 2.2.5, I can't find any proofs of it there.