nLab cartesian object (changes)

Proof

The idea of the proof is to reduce to the case of Cat using the 2-categorical Yoneda embedding.

Let $X$ be a cartesian object in $C$, and make a choice of right adjoints to the diagonal and terminal maps. Because 2-functors preserve adjunctions and the 2-Yoneda embedding $C \to [C^{op}, Cat]$, $A \mapsto C(-,A)$ preserves 2-products, $C(-,X)$ is a cartesian object with a choice of right adjoints in the 2-category of 2-presheaves on $C$. In particular, since 2-products in this 2-category are computed pointwise, $C(Y,X)$ is a cartesian object with a choice of right adjoints in Cat, and hence has the structure of a symmetric pseudomonoid in Cat, for each object $Y$ of $C$. Then, by similar reasoning, $C(-,X)$ is a symmetric pseudomonoid in the 2-presheaf 2-category. Finally, because the 2-Yoneda embedding is locally fully faithful and in particular locally reflects isomorphisms, $X$ is a symmetric pseudomonoid in $C$.

For any other choice of rights adjoints for the cartesian object $X$, there are unique invertible 2-cells between them that commute with the units and counits of the adjunctions. Threading these through the 2-Yoneda embedding as before, the 2-cells define the comparison iso-cells of a strong morphism of symmetric pseudomonoids whose underlying morphism between the carrier objects is the identity.