Proof that the left or right adjoint composed after a diagonal morphism in a 2-category is the identity

Question

Suppose I have some strict 2-category $\mathcal{C}$ that's cotensored over $\mathcal{C}at$. Then for $A\in \mathcal{C}$ and $X\in \mathcal{C}at$, there's the diagonal morphism $\Delta:A\to A^X$. Suppose $\Delta$ has a left adjoint $\Delta_!$. I want to show that $\Delta_!\Delta\cong\mathrm{id}_A$. More specifically, I expect that that 2-isomorphism is exhibited by the counit of the adjunction. Is this always true (as it obviously is when $\mathcal{C}=\mathcal{C}at$), and if so, how is it proven?

Answer 1

Just work representably: by the universal property of the cotensor and the fact that 2-functors preserve adjunctions, the sequence you get upon homming some $R$ into this is $A^R\to \mathrm{Cat}(X,A^R)\to A^R$ where the maps are the diagonal and the colimit.