# Kerodon

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Example 4.4.4.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, and suppose we are given a pair of diagrams $f,f': X \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then a natural transformation from $f$ to $f'$ can be identified with a collection of morphisms $\{ u_{x}: f(x) \rightarrow f'(x) \} _{x \in X}$ with the following property: for every edge $e: x \rightarrow y$ of the simplicial set $X$, the diagram

$\xymatrix@R =50pt@C=50pt{ f(x) \ar [r]^-{ u_ x } \ar [d]^{ f(e) } & f'(x) \ar [d]^{ f'(e) } \\ f(y) \ar [r]^-{u_ y} & f'(y) }$

commutes (in the category $\operatorname{\mathcal{C}}$).

In particular, if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are ordinary categories and we are given a pair of functors $f,f': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, then giving a natural transformation from $f$ to $f'$ (in the sense of classical category theory) is equivalent to giving a natural transformation from $\operatorname{N}_{\bullet }(f): \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ to $\operatorname{N}_{\bullet }(f'): \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.