Natural transformations and the definition of Monoidal lax functors The definition of a lax monoidal functor requires the existence of a natural transformation, $\phi$ http://en.wikipedia.org/wiki/Monoidal_functor. 


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*A natural transformation relates at least 2 functors; does this mean a lax monoidal functor is actually a collection of functors? If so, how can this collection be described? For example, is there a different functor for each object in the category $\mathcal{C}$ (i.e. if $\mathcal{C}=\{A,B,C...\}$,then $F=\{F_A,F_B,F_C,...\}$)?   

*The natural transformation is defined as $\phi_{A,B}:FA\bullet FB\to F(A\otimes B)$. $\phi$'s definition contains $F$ thrice so is it relating three different kinds of $F$? I was expecting $\phi$'s definition to prominently feature two different functors, instead I see three identical ones...? 

*Why is $\phi$ defined with index $\{A,B\}$. Does this mean there is actually a family of natural transformations $\phi=\{\phi_{A,B},\phi_{A,C},\phi_{B,C},...\}$?
Thanks!
 A: I hope what follows will clear up your confusion.
A lax monoidal functor consists of the following data (satisfying some axioms that I won't spell out):


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*One functor $\color{red}{F : \mathsf{C} \to \mathsf{D}}$. This means that for all objects $A \in \mathsf{C}$ you have an object $F(A) \in \mathsf{D}$, and for all morphisms $f : A \to B$ in $\mathsf{C}$ you have a morphism $F(f) : F(A) \to F(B)$ in $\mathsf{D}$. All this needs to satisfy some axioms.

*From this functor, you can construct two different functors $\mathsf{C} \times \mathsf{C} \to \mathsf{D}$, where $\mathsf{C} \times \mathsf{C}$ is the product of the category $\mathsf{C}$ with itself:


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*A functor $G_1$ that maps $(A,B)$ to $F(A \otimes B)$;

*A functor $G_2$ that maps $(A,B)$ to $F(A) \mathbin\bullet F(B)$.


Then you want one natural transformation $\color{red}{\phi : G_1 \to G_2}$ between these two functors. This means that for every object $(A,B)$ of $\mathsf{C} \times \mathsf{C}$, you have the data of a morphism $\phi_{(A,B)} : F(A \otimes B) \to F(A) \mathbin\bullet F(B)$, again satisfying some axioms.
Every occurrence of $F$ represent the same functor, but you construct other functors out of it. $\phi$ is indexed by objects of $\mathsf{C} \times \mathsf{C}$, ie. pairs $(A,B)$ of objects of $\mathsf{C}$. It does relate two different functors, $G_1$ and $G_2$, which are constructed out of $F$, $\otimes$ and $\bullet$.
