Kerodon

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

Example 2.1.6.6. Let $M$ and $M'$ be monoids, regarded as monoidal categories having only identity morphisms (Example 2.1.2.8). Then lax monoidal functors from $M$ to $M'$ (in the sense of Definition 2.1.5.8) can be identified with monoid homomorphisms from $M$ to $M'$ (in the sense of Definition 2.1.0.5). Moreover, every lax monoidal functor from $M$ to $M'$ is automatically strict monoidal (and therefore monoidal).