nLab cartesian object

A cartesian object in a 2-category with finite products is an object AA such that the diagonal morphism AA×AA\to A\times A and the unique map A1A\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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