I'm wondering whether there is an infinity categorical analogue to the results of Two-dimensional monad theory. For the most part, I'm interested in the relation between strict functors of infinity categories (with algebraic structure) and non-strict functors.

For example, it follows from the aforementioned paper that for every finitely complete category $\mathcal{C}$ there is a lex functor $I : \mathcal{C} \rightarrow \mathcal{C}'$ such that for every lex functor $J : \mathcal{C} \rightarrow \mathcal{D}$ there is a unique strict lex functor $K : \mathcal{C}' \rightarrow \mathcal{D}$ such that $J = K I$. (We assume that every finitely complete category comes with canonical choices of finite limits; strict lex functors need to preserve these canonical choices on the nose.) $I$ is an equivalence of categories, and $\mathcal{C}'$ is given by forgetting canonical limits in $\mathcal{C}$ and adjoining new canonical limits. The adjunction extends to a 2-adjunction, i.e. we get an isomorphism of categories $\mathrm{Lex}(\mathcal{C}, \mathcal{D}) \cong \mathrm{sLex}(\mathcal{C}', \mathcal{D})$.

Universal properties in the infinity categorical sense are defined via filling conditions, e.g. for inner horns in the case of composition. I suppose the obvious analogue of strict lex 1-functors would be a natural transformation of simplicial sets that preserves the canonical fillings to inner horns and the canonical fillers from the canonical limits in lex infinity categories.

I expect that the 1-categorical result I explained above also holds in the infinity categorical case. Has this (or something similar) been worked out somewhere?