Franklin's Notes

Diagonal functor

Given a functor category $\mathsf D^\mathsf C$, there is a diagonal functor $\Delta:\mathsf D\to\mathsf D^\mathsf C$ which sends each object $x\in\mathsf D$ to the constant functor $\Delta_x:\mathsf C\to\mathsf D$, and each morphism $f:x\to y$ of $\mathsf D$ to the corresponding natural transformation $\alpha:\Delta_x\Rightarrow\Delta_y$ whose only component is $f$. As a special case, consider the diagonal functor $\Delta:\mathsf C\to \mathsf C\times\mathsf C$ which acts on morphisms of $\mathsf C$ by sending $f\mapsto \langle f,f\rangle$. (This is, in fact, a special case of the above construction because $\mathsf C\times\mathsf C$ is isomorphic to the functor category $\mathsf C^{\cdot\cdot}$, where $\cdot\cdot$ represents the discrete category with two objects.)



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