[00KI]
Lemma B.2.3.
Let \(\mathcal C\) be an \(\infty\)-category equipped with a factorization system \((\mathcal L, \mathcal R)\), and let \(\mathcal I\) be a small \(\infty\)-category.
The \(\infty\)-category \(\mathrm{Fun}(\mathcal I, \mathcal C)\) admits a factorization system \((\mathcal L^\mathcal I,\mathcal R^\mathcal I)\), in which (as the exponential notation suggests) a natural transformation between functors lies in \(\mathcal L^\mathcal I\) (resp. \(\mathcal R^\mathcal I\)) if and only if its components all lie in \(\mathcal L\) (resp. \(\mathcal R\)).
If \(\mathcal C\) is presentable and \((\mathcal L,\mathcal R)\) is of small generation, then \(\mathrm{Fun}(\mathcal I,\mathcal C)\) is presentable and \((\mathcal L^\mathcal I,\mathcal R^\mathcal I)\) is of small generation.
If \(\mathcal O\) is a small operad, \(I\) is \(\mathcal O\)-monoidal, \(\mathcal C\) is presentably \(\mathcal O\)-monoidal and \((\mathcal L, \mathcal R)\) is compatible with the \(\mathcal O\)-monoidal structure on \(\mathcal C\), then \((\mathcal L^I, \mathcal R^I)\) is compatible with the Day convolution \(\mathcal O\)-monoidal structure on \(\mathrm{Fun}(I, \mathcal C)\).
[00KM]
Proof.
Part ([00KJ]) is a restatement of [Lur09, Cor. 5.2.8.18]. To prove part ([00KK]), assume that \(\mathcal C\) is presentable and let \(S\) be a set of morphisms in \(\mathcal C\) that generates \(\mathcal L\). We note first that \(\mathrm{Fun}(\mathcal I,\mathcal C)\) is presentable by [Lur09, Prop. 5.5.3.6]. Now, for each functor \({\sf pt}\xrightarrow{i} \mathcal I\) (selecting an object of \(\mathcal I\)) we obtain an adjunction 
It follows that \(\mathcal R^\mathcal I\) is precisely the right orthogonal to the (small) space of morphisms \[S'
\coloneqq
\bigsqcup_{i \in \iota_0 \mathcal I}
\bigsqcup_{f \in S}
i_!(f)\] in \(\mathrm{Fun}(\mathcal I,\mathcal C)\). From here, Proposition B.1.14 implies that \((\mathcal L^\mathcal I,\mathcal R^\mathcal I)\) is generated by \(S'\) (and in particular that \(\mathcal L^\mathcal I\) is the smallest saturated class of morphisms containing \(S'\)).
For part ([00KL]), given an n-ary operation \((X_1, \ldots, X_n) \rightarrow X\) in \(\mathcal O\), the induced functor \(\mathrm{Fun}(I_{X_1}, \mathcal C_{X_1}) \times \cdots \times \mathrm{Fun}(I_{X_n}, \mathcal C_{X_n}) \rightarrow\mathrm{Fun}(I_X, \mathcal C_X)\) is computed as the left Kan extension of \(I_{X_1} \times \cdots \times I_{X_n} \rightarrow\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_X\) against \(I_{X_1}\times \cdots \times I_{X_n} \rightarrow I_X\). As the left class of a factorization system, \(\mathcal L\) is closed under colimits in \(\mathcal C\). Therefore, the image of natural transformations \(f_i \in \mathrm{Fun}(I_{X_i}, \mathcal C_{X_i})\) which are componentwise in \(\mathcal L\) will again be componentwise in \(\mathcal L\). ◻