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Let $F,G:C\rightarrow D$ be two functors between categories. A natural transformation $\theta$ between $F$ and $G$ consists of the choice of a morphism $\theta_x$ for every $x \in \operatorname{Obj}(C)$ such that the wanted commutativity relations are satisfied.

Originally my question was: why do we choose morphisms for every object in $\operatorname{Obj}(C)$ and not in $\operatorname{Obj}(D)$? After thinking a bit about examples in homology it became clearer, that this seems just to be the thing we want. So my new question is: Is there a relevant concept of "natural transformations" where we choose morphisms for every object in the target-category?

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No, there is not such a concept.

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