Warning 8.4.3.4. The conclusion of Theorem 8.4.3.3 is not necessarily satisfied if we assume only that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small. For example, suppose that $\operatorname{\mathcal{C}}= S$ is a set of cardinality $\kappa $ (regarded as a discrete simplicial set), and let $\operatorname{\mathcal{D}}$ be (the nerve of) the partially ordered set $\{ 0 < 1 \} $. Then we can identify objects of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ with collections of $\kappa $-small Kan complexes $\{ X_ s \} _{s \in S}$. Define a functor $\lambda : \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ by the formula
and let $\lambda _0: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ be the constant functor taking the value $0$. The functors $\lambda $ and $\lambda _{0}$ both preserve $\kappa $-small colimits and coincide on the image of the Yoneda embedding $h_{\bullet }$, but do not coincide in general.