2-natural transformation?
symmetric monoidal (∞,1)-category of spectra
Doctrinal Adjunction is the title of a 1974 paper (Kelly) that gives conditions under which adjoint morphisms $f \dashv u$ in a 2-category $K$, and additionally the unit and counit, may be lifted to the category $T$-$Alg_l$ for some 2-monad $T$ on $K$.
Here $T$-$Alg_l$ is the 2-category of strict $T$-algebras, lax T-morphisms, and $T$-transformations, but the result works as well for pseudo algebras.
Let $f \dashv u$ be an adjunction in some 2-category $K$ and let $T$ be a 2-monad on $K$.
There is a bijection between 2-morphisms $\bar u$ making $(u,\bar u)$ a lax $T$-morphism and 2-morphisms $\tilde f$ making $(f,\tilde f)$ a colax $T$-morphism; it is given by taking mates with respect to the adjunctions $T f \dashv T u$ and $f \dashv u$.
The proof (Kelly) relies solely on the properties of the mate correspondence.
For the unit and counit of the adjunction $f \dashv u$ to be $T$-transformations, and hence for the adjunction to live in $T$-$Alg_l$, it is necessary and sufficient that $\tilde f$ have an inverse $\bar f$ that makes $(f,\bar f)$ into a lax $T$-morphism, and hence $(f,\bar f)$ into a strong $T$-morphism.
Again, the proof hinges on the properties of mates: we take the conditions for the unit and counit to be $T$-transformations and pass to mates wrt $T f \dashv T u$ and $1 \dashv 1$. Noting that $\tilde f$ is the mate of $\bar u$, the conditions are seen to be equivalent to requiring that $\bar f$ and $\tilde f$ are mutually inverse.
It follows that
$(f, \bar f) \dashv (u, \bar u)$ in $T$-$Alg_l$ if and only if $f \dashv u$ in $K$ and $\bar f$ has inverse $\tilde f$ = the mate of $\bar u$.
Doctrinal adjunction can be stated cleanly in terms of double categories. Namely, for any 2-monad $T$ there is a double category $T$-Alg whose objects are $T$-algebras, whose horizontal arrows are lax $T$-morphisms, whose vertical arrows are colax $T$-morphisms, and whose 2-cells are 2-cells in the base 2-category $K$ that make a certain cube commute. The horizontal 2-category of this double category is $T$-$Alg_l$, and its vertical 2-category is $T$-$Alg_c$. There is an obvious forgetful double functor $T \mathbf{Alg} \to \mathbf{Sq}(K)$, where $\mathbf{Sq}(K)$ is the double category of squares or “quintets” in $K$.
It is straightforward to verify that a conjunction in the double category $T$-Alg is precisely an adjunction in $K$ between $T$-algebras whose left adjoint is colax, whose right adjoint is lax, and for which the lax and colax structure maps are mates under the adjunction – i.e. a “doctrinal adjunction” in the above sense. Furthermore, an arrow in $T$-Alg has a companion precisely when it is a strong (= pseudo) $T$-morphism. The two central results of Kelly’s paper can then be stated as:
The forgetful double functor $U\colon T \mathbf{Alg} \to \mathbf{Sq}(K)$ creates conjunctions. I.e. given a horizontal arrow $u$ in $T \mathbf{Alg}$ and a left conjoint $f$ of $U(u)$ (i.e. a left adjoint of $u$ in $K$), there is a unique left conjoint of $u$ in $T\mathbf{Alg}$ lying over $f$.
Let $f\colon A\to B$ be a vertical arrow in $T \mathbf{Alg}$ (i.e. a colax $T$-morphism) and let $f'\colon A\to B$ and $u\colon B\to A$ be horizontal arrows (i.e. lax $T$-morphisms). Then from any two of the following three data we can uniquely construct the third.
Of these, the second is actually a general statement about companions and conjoints in any double category. Of course, the first is a special property of the forgetful double functor from the double category of $T$-algebras.
Let $K =$ Cat and $T$ the 2-monad whose 2-algebras are monoidal categories. Then
a lax $T$-morphism is a lax monoidal functor;
an oplax $T$-morphism is an oplax monoidal functor.
The above theorem then asserts
For two adjoint functors $(L \dashv R)$ between monoidal categories, $L$ is oplax monoidal precisely if $R$ is lax monoidal.
See oplax monoidal functor for more details.