# $2$-morphisms2-morphisms in structured $2$-categories2-categories
There are many $2$-subcategories of $Cat$, 2$-categories, which are first specified by their objectscertain 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, althouh 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". 2 edited body There are many$2$-subcategories of$Cat$, which are first specified by their objects; 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, althouh 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 wond 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".