Kerodon

Proof. Choose a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Fix objects $\widehat{X},\widehat{Y} \in \widehat{\operatorname{\mathcal{C}}}$; we wish to show that the morphism space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small. Let us first regard the object $\widehat{Y}$ as fixed, and let $\widehat{\operatorname{\mathcal{C}}}_0$ denote the full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by those objects $\widehat{X}$ for which the morphism space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small. Since $\lambda $ has exponential cofinality $\geq \kappa $, the collection of $\lambda $-small spaces is closed under formation of $\kappa $-small limits (Variant 7.4.1.4). It follows that $\widehat{\operatorname{\mathcal{C}}}$ is closed under $\kappa $-small colimits in $\widehat{\operatorname{\mathcal{C}}}$ (see Proposition 7.4.1.18). In particular, it is closed under $K$-indexed colimits for each $K \in \mathbb {K}$. Consequently, to show that $\widehat{\operatorname{\mathcal{C}}}_0 = \widehat{\operatorname{\mathcal{C}}}$, it will suffice to show that the mapping space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small in the special case $\widehat{X} = h(X)$ for some object $X \in \operatorname{\mathcal{C}}$.

Let us now regard the object $\widehat{X} = h(X)$ as fixed, and let $\widehat{\operatorname{\mathcal{C}}}_1$ denote the full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by those objects $\widehat{Y}$ for which the morphism space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, \widehat{Y})$ is essentially $\lambda $-small. For each $K \in \mathbb {K}$, the $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( h(X), \bullet )$ preserves $\mathbb {K}$-indexed colimits. Since the collection of $\lambda $-small spaces is closed under $K$-indexed colimits (Corollary 7.4.3.8), it follows that the subcategory $\widehat{\operatorname{\mathcal{C}}}_{1} \subseteq \widehat{\operatorname{\mathcal{C}}}$ is closed under $K$-indexed colimits. Consequently, to prove that $\widehat{\operatorname{\mathcal{C}}}_{1} = \widehat{\operatorname{\mathcal{C}}}$, it will suffice to show that it contains every object of the form $h(Y)$, for $Y \in \operatorname{\mathcal{C}}$. Since $h$ is fully faithful, we are reduced to proving that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is essentially $\lambda $-small, which follows from our assumption that $\operatorname{\mathcal{C}}$ is locally $\lambda $-small. $\square$