Diagrams in $(\infty,n)$-categories

Question

When working with homotopy coherent diagrams in an $(\infty,1)$-category $\mathcal{C}$ (viewing $(\infty,1)$-categories as quasi-categories), we can make sense of them as objects in $\operatorname{Map}_{\mathsf{Set}^{\Delta^\textrm{op}}}(K,\mathcal{C}),$ which is itself a quasi-category. For example, homotopy coherent squares in $\mathcal{C}$ are objects of $\operatorname{Map}_{\mathsf{Set}^{\Delta^\textrm{op}}}(\Delta^1\times\Delta^1,\mathcal{C}).$

My Question: Is there a similar way to obtain an $(\infty,n)$-category of homotopy coherent diagrams of shape $K\in\mathsf{Set}^{\Delta^\textrm{op}}$ in a given $(\infty,n)$-category? I would be happy with an answer for $n = 2,$ and for any particular model for $(\infty,n)$-categories.

Motivation: I'm currently reading "A toy model for the Drinfeld-Lafforgue shtuka construction," and there are a few places where arguments are made using diagrams in $(\infty,2)$ and $(\infty,3)$-categories, and I'm trying to understand these arguments a bit better. I've managed to translate much of the argument I'm particularly interested in into statements about functors between particular diagram categories in $(\infty,n)$-categories, but I haven't managed to find anything in the literature which formalizes this notion.

Answer 1

I think the answer I want is given by Johnson-Freyd and Scheimbauer's paper "(Op)lax natural transformations, twisted quantum field theories, and 'even higher' Morita categories". Here is a summary for anyone else who may need such notions.

For two $(\infty,n)$-categories $\mathcal{B}$ and $\mathcal{C},$ the authors construct three $(\infty,n)$-categories of functors from $\mathcal{B}$ to $\mathcal{C}.$

The first of these categories is $\operatorname{Fun}^\textrm{strong}(\mathcal{B},\mathcal{C}),$ which is determined by Cartesian closedness of $\mathsf{Cat}_{(\infty,n)}.$ That is, this category is determined by the universal property $$ \operatorname{Map}_{\mathsf{Cat}_{(\infty,n)}}(\mathcal{A},\operatorname{Fun}^\textrm{strong}(\mathcal{B},\mathcal{C}))\simeq\operatorname{Map}_{\mathsf{Cat}_{(\infty,n)}}(\mathcal{A}\times\mathcal{B},\mathcal{C}). $$

The second and third categories, $\operatorname{Fun}^\textrm{lax}(\mathcal{B},\mathcal{C})$ and $\operatorname{Fun}^\textrm{oplax}(\mathcal{B},\mathcal{C}),$ have the same space of objects as $\operatorname{Fun}^\textrm{strong}(\mathcal{B},\mathcal{C})$ (the "strong" functors $\mathcal{B}\to\mathcal{C}$), but the higher morphism spaces differ. In particular, in $\operatorname{Fun}^\textrm{strong}(\mathcal{B},\mathcal{C})$ the morphisms between diagrams must commute up to isomorphism, but in $\operatorname{Fun}^\textrm{lax}(\mathcal{B},\mathcal{C})$ and $\operatorname{Fun}^\textrm{oplax}(\mathcal{B},\mathcal{C}),$ morphisms between diagrams must only commute up to specified natural transformation (and the direction of this natural transformation is determined by whether we use the lax or oplax variant).

These are constructed using "$n$-computads," which are roughly presentations of free strict $n$-categories by generating families of morphisms. As an example, $\Theta^{(1)}$ represents the "walking $1$-morphism" $\bullet\to\bullet.$ Using this computad, the authors construct $[\Theta^{(1)},\mathcal{C}],$ $\mathcal{C}^{\downarrow},$ and $\mathcal{C}^{\rightarrow},$ which are essentially three incarnations of the arrow category of $\mathcal{C}.$ The space of objects of each are maps $c_1\to c_2$ in $\mathcal{C},$ but maps between $(c_1\to c_2)$ and $(d_1\to d_2)$ in $[\Theta^{(1)},\mathcal{C}]$ are squares which commute up to isomorphism, maps between $(c_1\to c_2)$ and $(d_1\to d_2)$ in $\mathcal{C}^{\downarrow}$ are squares which commute up to a natural transformation $\eta : (c_1\to c_2\to d_2)\implies (c_1\to d_1\to d_2),$ and maps between $(c_1\to c_2)$ and $(d_1\to d_2)$ in $\mathcal{C}^{\rightarrow}$ are squares which commute up to a natural transformation $\eta : (c_1\to d_1\to d_2)\implies (c_1\to c_2\to d_2).$

One question which I did not see an answer to in the paper is whether the lax and oplax categories of functors satisfy universal properties analogous to the universal property for the strong category. I imagine this might be the case when we replace $\times$ by the Gray tensor product, but I don't know enough about these things to say. (Perhaps the answer to this is obvious to experts or I just missed it in the paper -- I would love to hear if this is indeed the case!)