$\operatorname{Fun}(\mathcal{C},\mathcal{D})^n$ is a subcategory of $\operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n)$

Question

Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories (by which I mean quasicategories, though I suspect that it hardly matters), and let $n\geq 1$ be an integer. There is a functor

$$\theta:\operatorname{Fun}(\mathcal{C},\mathcal{D})^n\to\operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n)$$

given by $(F_1,\dots ,F_n)\mapsto F_1\times \cdots \times F_n$. Is there a subcatetgory $\mathcal{X}\subset \operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n)$ such that $\theta$ restricts to a categorical equivalence $\operatorname{Fun}(\mathcal{C},\mathcal{D})^n\xrightarrow{\simeq}\mathcal{X}$? For ordinary categories, this is a trivial question, but I do not know how to address this problem for $\infty$-categories.

Essentially, what I am asking is whether $\theta$ is homotopically faithful, in the sense that given functors $F_1,\dots ,F_n,G_1,\dots, G_n:\mathcal{C}\to \mathcal{D}$, the map

$$\prod_{i=1}^n\operatorname{Hom}_{\operatorname{Fun}(\mathcal{C},\mathcal{D})}(F_i,G_i) \to \operatorname{Hom}_{\operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n)}(\prod_i F_i, \prod_i G_i)$$

can be identified with the inclusion of components (i.e., injective in $\pi_0$ and bijective in $\pi_i$ for $i\geq 1$). Does anyone know how to prove this? Thanks in advance.

Tangential Remark

I ran into this problem while I was reading Saul Glasman's paper "Day convolution for $\infty$-categories." There (right after Definition 2.8) he seems to claim that we can take $\mathcal{X}$ as the following subcategory of $\operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n)$:

• Objects of $\mathcal{X}$ are the objects lying in the essential image of $\theta$.
• Morphisms of $\mathcal{X}$ are those lying in the essential image of the functor $\operatorname{Fun}([1],\operatorname{Fun}(\mathcal{C},\mathcal{D}))^n\to \operatorname{Fun}([1],\operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n))$.

Unfortunately, this does not seem to work well. (For example, we can take $\mathcal{C}$ as the nerve of any discrete category containing at least two objects, and $\mathcal{D}$ as the nerve of any category containing an object with a nontrivial automorphism; in this case we can check that $\theta:\operatorname{Fun}(\mathcal{C},\mathcal{D})^n\to\mathcal{X}$ is not full.)

Answer 1

Note that $Fun(C^n,D^n) = Fun(C^n,D)^n$, so your second mapping space is also a product of mapping spaces.

In most cases (i.e. except when one of the terms is empty), a product of maps is an inclusion of components if and only if each of terms is. In particular, your question is often equivalent to (or more precisely, the following is a sufficient condition which is "essentially necessary", though not quite) each of the $\pi_i^*: Fun(C,D) \to Fun(C^n, D)$ being a subcategory inclusion, i.e. each of the $\pi_i: C^n\to C$ being an epimorphism (if we let $D$ be arbitrary).

In turn, if $C$ is non-empty, this is equivalent to the map $C^{n-1}\to *$ being an epimorphism. The following is an elementary lemma:

Lemma: $E\to *$ is an epimorphism if and only if $|E|\to *$ is, if and only $\Sigma |E|\simeq *$.

In particular, the following is a concretely checkable criterion to get a positive answer:

Corollary: If $|C| \simeq *$, then $Fun(C,D)^n\to Fun(C^n,D^n)$ is a faithful category inclusion.

The proof (and the second paragraph of this answer) also suggest that "$\Sigma |C|\simeq *$" is not far from a necessary condition.

It also suggests some counterexamples. For example:

Example: Let $C= X$ be a space/$\infty$-groupoid, and $D = K(A,k)$ for some $k\geq 1$ and some abelian group $A$. The comparison map is, on $\pi_j$, $$H^{k-j}(X;A)^n\to H^{k-j}(X^n,A)^n, (x_i)_{i\leq n}\mapsto (\pi_i^* x_i)_i$$.

This has no reason to be an isomorphism for all $j\geq 2$ (e.g. take $k\geq 3$).