Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.1.1.10 (Nonunital Monoidal Structures on Functor Categories). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. Then every nonunital monoidal structure $(\otimes , \alpha )$ on $\operatorname{\mathcal{D}}$ determines a nonunital monoidal structure on the functor category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, whose underlying tensor product is given by the composition

\[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}\times \operatorname{\mathcal{D}}) \xrightarrow { \otimes \circ } \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]

and whose associativity constraint assigns to each triple of functors $F,G,H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ the natural isomorphism

\[ F \otimes (G \otimes H) \xrightarrow {\sim } (F \otimes G) \otimes H \quad \quad C \mapsto \alpha _{ F(C), G(C), H(C)}. \]