nLab cartesian object

2-category theory

Structures on 2-categories

A cartesian object in a 2-category with finite products is an object $A$ such that the diagonal morphism $A\to A\times A$ and the unique map $A\to 1$ have right adjoints. Any cartesian object is automatically a pseudomonoid in a canonical way.

For example, a cartesian object in Cat is precisely a category with finite products, which is of course a monoidal category in a canonical way.

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