Functors preserving coproducts first variable

Notation 15.2.11. Let \(C\), \(D\) and \(E\) be \(\infty \)-categories and assume \(D\) and \(E\) admit finite coproducts. We denote by \[ \Fun ^{\amalg ,-}(E \times C,D) \quad \subseteq \quad \Fun (E \times C, D) \] the full subcategory consisting of functors \(F\colon E \times C \to D\) that preserve finite coproducts in the first variable, in the sense that \(F(-,x)\colon E \to D\) preserves finite coproducts for all \(x \in C\).