Natural transformation

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Definition

Suppose ${\mathcal {C}},{\mathcal {D}}$ are categories, and ${\mathcal {F}}:{\mathcal {C}}\to {\mathcal {D}}$ and ${\mathcal {G}}:{\mathcal {C}}\to {\mathcal {D}}$ are functors. A natural transformation $\eta :{\mathcal {F}}\to {\mathcal {G}}$ is defined by the following data:

For every object $A\in \operatorname {Ob} {\mathcal {C}}$, a morphism $\eta _{A}\in {\mathcal {D}}({\mathcal {F}}A,{\mathcal {G}}A)$,

satisfying the following compatibility condition: For any objects $A,B\in \operatorname {Ob} {\mathcal {C}}$, and any $f\in {\mathcal {C}}(A,B)$, we have:

${\mathcal {G}}f\circ \eta _{A}=\eta _{B}\circ {\mathcal {F}}f$.