In Day–Street's Lax monoids, pseudo-operads, and convolution, they remark without proof:
There are general principles involved here. Suppose $(T, m, j)$ is a pseudomonad on any bicategory $\mathcal K$. If $m : TT \to T$ and $j : 1 \to T$ have right adjoints $m^*$, $j^*$ then we obtain a pseudocomonad $(T, j^*, m^*)$ on $\mathcal K$; moreover, to give a lax algebra for $(T, m, j)$ is to give a monad in the Kleisli bicategory for $(T, j^*, m^*)$.
Is there a reference for this result? Alternatively, if it is folklore, is there a slick proof that avoids having to check the coherence diagrams explicitly?
