TITLE: 2-morphisms in structured 2-categories
QUESTION [7 upvotes]: There are many $2$-categories, which are first specified by certain categories with extra structure; then the $1$- and $2$-morphisms are functors and natural transformations that preserve the extra structure. I want to understand the general procedure in finding the "correct" definitions of these $2$-morphisms, if there is any.
Example 1: Objects are tensor categories. Then $1$-morphisms should be tensor functors (some allow them to be lax) and $2$-morphisms are natural transformations $\eta$ which are compatible with the tensor structure. This means that $\eta(1)$ is an isomorphism and that for every pair of objects $x,y$ we have a commutative diagram which identifies $\eta_{x \otimes y}$ with $\eta_x \otimes \eta_y$.
Example 2: Take as objects cocomplete categories. Then $1$-morphisms are cocontinuous functors and $2$-morphisms are natural transformations $\eta$ which preserve colimits. The latter means that that for every colimit $\colim_i x_i$ the morphism $\eta(x)$ is the colimit of the morphisms $\eta(x_i)$. But wait, this is automatically true! This follows easily from the cocontinuity of the functors and the naturality of $\eta$. In how far is this "coincidence"?
So far I have never seen this definition of a "cocontinuous natural transformation", but actually this property is used very often when dealing with natural transformations in this situation. So perhaps it should be included in the definition? For example the "correct" definition of a homomorphism $f : G \to H$ of groups includes that $f$ preserves the unit, inversion and multiplication, although everyone knows that multiplication is enough and unfortunately some authors then take the "wrong" definition and get the correct one by a lemma. I hope it's clear that I don't want to offend anyone here and there is no "correct" definition, but perhaps the one which fits best into general patters of category theory.
Example 3: Objects are symmetric tensor categories. Then $1$-morphisms are tensor functors which preserve the symmetry (the functor $F$ maps the symmetry $x \otimes y \cong y \otimes x$ to the symmetry $F(x) \otimes F(y) \cong F(y) \otimes F(x)$; again this is a commutative diagram) and $2$-morphisms are natural transformations $\eta$ which are compatible with the tensor structure as in Example 1 and also are compatible with the symmetry. But what should this compatibility mean? Actually I have not been able to write down a diagram which connects $\eta$ with the symmetry and does not directly follow from the naturality. So perhaps we cannot even formulate a compatibiltiy condition here? Again I'm interested in how far this is "coincidence".

REPLY [4 votes]: A. This is really just an aspect of Mike Shulman's answer, but could be of some use in particular cases.
There's a 2-categorical limit called the power (or cotensor) of an object $B$ by the arrow-category $2$. This is an object $B^2$ with the property that morphisms from $A$ to $B^2$ are in bijection with pairs of morphism from A to B with a 2-cell between them. For example
if B is a category then $B^2$ is the functor category $[2,B]$. If $B$ is a monoidal category then $B^2$ is $[2,B]$ with the evident (pointwise) monoidal structure.
In each of your examples, and more generally in Mike's setting, this limit exists in the 
structured 2-category, and is preserved by the forgetful 2-functor into Cat. Normally you would prove this given the definition of 2-cell. But you can also turn this around. Given a 
structure on B, if you know how to make $B^2$ into a structured object, then you can use this to define the structured 2-cells. 
In examples where the structure is given by a 2-monad, and in particular in examples which
involve structure described by operations $B^n\to B$, natural transformations between these, and equations, then you can always do this in a "pointwise way".
(But if you choose a strange way to make $B^2$ into a structured object you will get a strange notion of 2-cell.)
Suppose, for example, that $B$ is a monoidal category. Once you agree to make $[2,B]$ monoidal in the pointwise way, then you can define a monoidal transformation to be a monoidal functor with codomain $[2,B]$, and this will agree with the standard definition which you referred to.
In the case of a cocomplete category $B$, you don't need to choose how to make $[2,B]$ 
cocomplete, it just is. And then you can consider cocontinuous functors with codomain $[2,B]$; once again this will give no extra condition to be satisfied by  a natural transformation between cocontinuous functors 
The case of symmetric monoidal categories can be treated in the same way.
B. Regarding the case of symmetric monoidal categories, there is a general phenomenon here. As you add structure to your objects in the form of operations $B^n\to B$ (like a tensor product) you generally introduce preservation conditions on both morphisms and 2-cells (although there are special cases, as in your Example 2, where the 2-cell part is automatic). But if you introduce structure in the form of natural transformations between the operations $B^n\to B$ (such as a symmetry), this results in new preservation conditions for the morphisms but not for the 2-cells. 
C. Despite all this, there can be more than one choice for the 2-cells. The general principles described by Mike (and by me) would suggest that if our structure is categories with pullback, so that our morphisms are pullback-preserving functors, the 2-cells should be all natural transformations between these. But sometimes it's good to consider only those natural transformations for which the naturality squares are pullbacks. (These are sometimes called cartesian natural transformations.) See this paper for example.