ScalingStacks

Definition 2.6.3.

For a monoidal functor \(F\colon \mathcal A\rightarrow\mathcal B\), we define the monoidal evaluation functor \[\mathrm{ev}\colon Z(F) \times \mathcal A\rightarrow\mathcal B\] to send an object \(((b,\gamma),a)\) to \(b\boxtimes F(a)\), and a morphism \((f,g)\) to \(f\boxtimes F(g)\). The monoidal structure isomorphisms \[\mathrm{ev}\left(((b,\gamma),a)\boxtimes((b',\gamma'),a')\right)\simeq \mathrm{ev}\left((b,\gamma),a\right)\boxtimes \mathrm{ev}\left((b',\gamma'),a')\right) \quad(\text{for } a,a'\in\mathcal A,(b,\gamma),(b',\gamma')\in Z(F))\] are given by \(b\boxtimes b'\boxtimes F(a\boxtimes a')\simeq b\boxtimes b'\boxtimes F(a)\boxtimes F(a') \xrightarrow{\mathrm{id}\boxtimes\gamma'_a\boxtimes\mathrm{id}} b\boxtimes F(a)\boxtimes b'\boxtimes F(a')\). The defining properties of \(\gamma'\) and the monoidality of \(F\) ensure that the necessary compatibilities hold, so that we indeed get a monoidal functor.