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$\kappa $-left-exact copresheaves on a $(\kappa ,\kappa )$-coherent category$\mathcal{C}$ form a full reflective subcategory in $\mathcal{C}\downarrow\mathbf{Coh}^{\sim }_{\kappa ,\kappa }$, the category of $(\kappa ,\kappa)$-coherent functors out of $\mathcal{C}$. Given a $\kappa $-lex$F:\mathcal{C}\to \mathbf{Set}$ the construction extends $\mathcal{C}$ byadding constants along $F$.  $\kappa $-regular copresheaves are identified with $(\kappa ,\kappa)$-coherent functors to sheaves over $\kappa $-complete, $(\kappa ,\kappa)$-distributive Boolean-algebras. As an application we prove an infinitaryversion of the Heyting-valued completeness theorem, and characterise which$\kappa $-filtered colimit preserving functors $\mathbf{Coh}_{\kappa ,\kappa}(\mathcal{D},\mathbf{Set})\to \mathbf{Coh}_{\kappa ,\kappa}(\mathcal{C},\mathbf{Set})$ are coming from an interpretation $\mathcal{C}\to\mathcal{D}$.  We investigate the functorial properties of two semiring invariants of lexcopresheaves over coherent categories.

$\kappa $-left-exact and $\kappa $-regular copresheaves on $(\kappa ,\kappa )$-coherent categories