We now record an important special case of Theorem 8.4.3.2 (compare with Example 8.4.0.4:
Theorem 8.4.4.1. Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete. If $X \in \operatorname{\mathcal{S}}_{< \kappa }$ is a contractible Kan complex, then evaluation at $X$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{S}}_{< \kappa }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad T \mapsto T(X). \]
Proof. Since $X$ is contractible, the identity functor $\operatorname{\mathcal{S}}_{< \kappa } \rightarrow \operatorname{\mathcal{S}}_{< \kappa }$ is corepresentable by $X$ (see Proposition 5.6.6.17), so that the inclusion functor
is a covariant Yoneda embedding for the standard $0$-simplex $\Delta ^0$. The desired result now follow from Theorem 8.4.3.2. $\square$
Remark 8.4.4.2 (Functoriality of Copowers). Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete. Theorem 8.4.4.1 guarantees that for every object $C \in \operatorname{\mathcal{C}}$, there is an essentially unique $\kappa $-cocontinuous functor $T: \operatorname{\mathcal{S}}_{< \kappa } \rightarrow \operatorname{\mathcal{C}}$ such that $T( \Delta ^0 )$ is isomorphic to $C$. Using Corollary 8.4.3.8, we see that this functor carries a $\kappa $-small Kan complex $K$ to a colimit of the constant diagram $K \rightarrow \{ C\} \hookrightarrow \operatorname{\mathcal{C}}$. Stated more informally, we have isomorphisms $T(K) \simeq K \otimes C$, where $K \otimes C$ is copower of $C$ by $K$ (see Definition 7.1.2.1 and Notation 7.1.2.5).
Corollary 8.4.4.3. Let $\kappa $ be an uncountable regular cardinal, let $n$ be an integer, and let $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa } \subseteq \operatorname{\mathcal{S}}_{< \kappa }$ be the full subcategory spanned by those Kan complexes which are $n$-truncated and essentially $\kappa $-small. Let $X \in \operatorname{\mathcal{S}}^{\leq n}_{< \kappa }$ be a contractible Kan complex and let $\operatorname{\mathcal{C}}$ be a $\kappa $-cocomplete $\infty $-category. Then evaluation at $X$ determines a fully faithful functor whose essential image is spanned by those objects $C \in \operatorname{\mathcal{C}}$ which are $n$-cotruncated (Variant 4.7.1.4).
\[ \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{S}}^{\leq n}_{< \kappa }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad T \mapsto T(X), \]
Remark 8.4.4.4. Corollary 9.5.5.12 has a counterpart in the case $\kappa = \aleph _0$, which is a bit more subtle to formulate. See Corollary 9.5.5.12.
Proof of Corollary 8.4.4.3. Recall that $\operatorname{\mathcal{S}}_{< \kappa }^{\leq n}$ is a reflective subcategory of $\operatorname{\mathcal{S}}_{< \kappa }$: that is, the inclusion functor $\operatorname{\mathcal{S}}_{< \kappa }^{\leq n} \hookrightarrow \operatorname{\mathcal{S}}_{< \kappa }$ admits a left adjoint $L$, given on objects by the construction $L(Y) = \pi _{\leq n}(Y)$ (see Variant 6.2.2.13). Applying Proposition 6.3.3.7 and Remark 7.1.4.32, we see that precomposition with $L$ induces a fully faithful functor
whose essential image is spanned by those $\kappa $-cocontinuous functors $T: \operatorname{\mathcal{S}}_{< \kappa } \rightarrow \operatorname{\mathcal{C}}$ which satisfy the following condition:
If $f: K \rightarrow K'$ is a morphism of essentially $\kappa $-small Kan complexes which induces a homotopy equivalence $\pi _{\leq n}(K) \rightarrow \pi _{\leq n}(K')$, then the induced map $T(f): T(K) \rightarrow T(K')$ is an isomorphism in $\operatorname{\mathcal{C}}$.
By virtue of Theorem 8.4.4.1, it will suffice to show that condition $(\ast )$ is equivalent to the following:
The functor $T$ carries the contractible Kan complex $X$ to an $n$-cotruncated object of $\operatorname{\mathcal{C}}$.
We first show that $(\ast ) \Rightarrow (\ast ')$. Assume that condition $(\ast )$ is satisfied. Then $T$ restricts to a $\kappa $-cocontinuous functor $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa } \rightarrow \operatorname{\mathcal{C}}$, which carries $n$-cotruncated objects of $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa }$ to $n$-cotruncated objects of $\operatorname{\mathcal{C}}$ (see Corollary 7.6.2.29). Since the $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa }$ is locally $n$-truncated (Example 4.7.1.8), the Kan complex $X$ is $n$-cotruncated when viewed as an object of $\operatorname{\mathcal{S}}^{\leq n}_{< \kappa }$, so its image $T(X) \in \operatorname{\mathcal{C}}$ is $n$-cotruncated.
We now prove the converse. Assume that $C = T(X)$ is an $n$-cotruncated object of $\operatorname{\mathcal{C}}$ and let $f: K \rightarrow K'$ be a morphism of essentially $\kappa $-small Kan complexes which induces a homotopy equivalence $\pi _{\leq n}(K) \rightarrow \pi _{\leq n}(K')$; we wish to show that $T(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Let $D \in \operatorname{\mathcal{D}}$ be an object; we wish to show that precomposition with $T(f)$ induces a homotopy equivalence of morphism spaces $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}( T(K'), D) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( T(K), D)$. Using Remark 8.4.4.2, we can identify $\theta $ with the map $\operatorname{Fun}(K', \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) ) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) )$ given by precomposition with $f$. The assumption that $C$ is $n$-cotruncated guarantees that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ is $n$-truncated, so that $\theta $ is a homotopy equivalence. $\square$
Corollary 8.4.4.5. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be a cocomplete $\infty $-category. If $X$ is a contractible Kan complex, then evaluation at $X$ induces a fully faithful functor whose essential image is spanned by the $n$-cotruncated objects of $\operatorname{\mathcal{C}}$.
\[ \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{S}}^{\leq n}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad T \mapsto T(X), \]
Proof. Apply Corollary 8.4.4.3 in the special case where $\kappa = \Omega $ is a strongly inaccessible cardinal. $\square$
Example 8.4.4.6. Let $\operatorname{\mathcal{C}}$ be a cocomplete $\infty $-category and let $\operatorname{Set}$ be the category of sets. Then the evaluation functor is fully faithful, and its essential image is spanned by those objects $C \in \operatorname{\mathcal{C}}$ which are codiscrete (that which are discrete when viewed as objects of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$). This follows by applying Corollary 8.4.4.5 in the special case $n = 0$ (see Example 4.7.4.17).
\[ \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{N}_{\bullet }(\operatorname{Set}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad T \mapsto T(\ast ) \]