Left derivable monoidal structure

Definition 20.1.3 (Left derivable monoidal structure). Let \((C,W,I)\) be an \(\infty \)-category with weak equivalences and cofibrations, and assume that \(C\) is given a symmetric monoidal structure. We call an object of \(C\) tensor-cofibrant if it belongs to the smallest full subcategory of \(C\) that contains all cofibrant objects and the monoidal unit and is closed under binary tensor products. We say that the monoidal structure is left derivable if for every finite set \(J\), every \(J\)-indexed collection of cofibrant objects \(\{X_j\}_{j \in J}\), every \(J\)-indexed collection of tensor-cofibrant objects \(\{Y_j\}_{j \in J}\), and every collection of weak equivalences \(f_j\colon X_j \to Y_j\), the induced map \[ \bigotimes _{j \in J} f_j \colon \bigotimes _{j \in J} X_j \to \bigotimes _{j \in J} Y_j \] is again a weak equivalence.