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Are the components of a natural transformation $ \alpha : F \rightarrow G $, where $F, G : C \rightarrow D$ are functors, always morphisms "of" $D$? or can we just"imagine" them?

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what do you mean by "just imagine them" ? in any case, as @andy explained below, the answer is YES, they are always morphisms of the target category D –  magma Jul 6 '12 at 14:03
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Since $\alpha_c \colon Fc \to Gc$ and $Fc,Gc$ are objects of $D$ (we take $c$ to be an object of $C$), then $\alpha_c$ needs to be a morphism of $D$. If it isn't but the diagram does commute, then it is not a natural transformation of $F,G$ as functors to $D$, but to a category in which $D$ embeds fully and faithfully. If it is, but the diagram doesn't commute, then it is not a natural transformation.

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