Given categories $\mathsf{C}$ and $\mathsf{D}$ and functors $F,G \colon \mathsf{C} \rightrightarrows \mathsf{D}$, a natural transformation $\alpha \colon F \Rightarrow G$ consists of:\n1. an arrow $\alpha_c \colon Fc \to Gc$ in $\mathsf{D}$ for each object $c\in \mathsf{C}$, the collection of which define the components of the natural transformation,\nso that, for any morphism $f \colon c \to c’$ in $\mathsf{C}$, the following square of morphisms in $\mathsf{D}$\n\(\vcenter{\xymatrix{ Fc \ar[d]_{Ff} \ar[r]^{\alpha_c} & Gc \ar[d]^{Gf} \\ Fc' \ar[r]_{\alpha_{c'}} & Gc'}}\)\ncommutes, i.e., has a common composite $Fc \to Gc’$ in $\mathsf{D}$.\n\nA natural isomorphism is a natural transformation $\alpha \colon F \Rightarrow G$ in which every component $\alpha_c$ is an isomorphism. In this case, the natural isomorphism may be depicted as $\alpha \colon F \cong G$.