certain horn filling conditions. In addition, we use the following notation, some of which differs from Lurie’s:

-   ${\mathbb{\Delta}}$ denotes the simplicial indexing category whose objects are non empty fnite totally ordered sets $[n] := \{0, 1, . . . , n\}$ and morphisms are order-preserving functions between them. $\Delta^n:{\mathbb{\Delta}}^{op}\to {\operatorname}{Sets}$ is the simplicial set represented by $[n]$.

-   We denote generic $\infty$-categories by upper-case caligraphic letters, ${\mathcal}{A},{\mathcal}{B},{\mathcal}{C},{\mathcal}{D},\dots$. We typically denote elements $c\in {\mathcal}{C}$ of a generic $\infty$-category by lowercase versions of the same letter.

-   We let ${\operatorname}{Fun}({\mathcal}{C},{\mathcal}{D})$ denote the $\infty$-category of functors between $\infty$-categories, and ${\operatorname}{Map}_{{\mathcal}{C}}(c,c')$ denote the mapping space between two objects $c,c'\in {\mathcal}{C}$.

-   If ${\mathcal}{C}$ is an $\infty$-category, we write $\iota{\mathcal}{C}$ for the *interior* or *classifying space of objects* of ${\mathcal}{C}$, i.e. the maximal Kan complex contained in ${\mathcal}{C}$.

-   If $f : {\mathcal}{C} \to {\mathcal}{D}$ is left adjoint to a functor $g : {\mathcal}{D} \to{\mathcal}{C}$, we will refer to the adjunction as $f \dashv g$.

-   ${{\mathscr{C}\mathrm{at}}}_\infty$ denotes the $\infty$-category of small $\infty$-categories, and the $\infty$-category of spaces, ${\mathscr{S}}\subset {{\mathscr{C}\mathrm{at}}}_\infty$, is the full subcategory spanned by the Kan complexes.

-   If ${\mathcal}{C}$ is an $\infty$-category, we let $${\!\text{\usefont{U}{min}{m}{n}\symbol{'210}}\!}:{\mathcal}{C}\to {\mathscr{P}}({\mathcal}{C}):={\operatorname}{Fun}({\mathcal}{C},{\mathscr{S}})$$ denote the Yoneda embedding[^3].

-   