monoid object

1. Definition

Let 20231110-monoid_object_760824120891fc757bc0445145b027d09b237baa.svg be a monoidal category and 20231110-monoid_object_d81849da0f4e3c91fdbd862726b462ee3137c8a1.svg . Then 20231110-monoid_object_612a3d41795adef416750fa250ce585eadd8340d.svg (with following morphisms) is said to be a monoid object, if

there exists a morphism for the tensor unit
there exists a morphism , called the multiplication functor
associativity holds, i.e.

20231110-monoid_object_5fa1d9ec89c217dfbe7048180a59fbb841765e08.svg

for the associator 20231110-monoid_object_9339d7e01d68ebe088aba117e0f285dd9c566c3a.svg

the diagram

20231110-monoid_object_aad555740fda573ad963fd2da17b5f7db5ebaa21.svg

commutes