Cone and cocone

A cone from an object $𝐴 \in 𝖢$ to a diagram $𝒟 : 𝖩 \to 𝖢$ is a natural transformation $𝜓 : 𝐴 \Rightarrow 𝒟 : 𝖩 \to 𝖢$ from the constant functor at $𝐴$ . #m/def/cat Hence for all $𝑖, 𝑗 \in 𝖩$ and $𝛼 \in 𝖩 𝑖, 𝑗$ the following diagram commutes in $𝖢$ :

c|https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzAsMSwiXFxtYXRoc2NyIERfaSJdLFsyLDEsIlxcbWF0aHNjciBEX2oiXSxbMSwyLCJcXG1hdGhzY3IgRF97XFxhbHBoYX0iXSxbMCwxLCJcXHBzaV9pIiwyXSxbMCwyLCJcXHBzaV9qIl1d

Dually, a cocone from a diagram $𝒟 : 𝖩 \to 𝖢$ to an object $𝐴 \in 𝖢$ is a natural transformation $𝜓 : 𝒟 \Rightarrow 𝐴 : 𝖩 \to 𝖢$ .

c|https://q.uiver.app/#q=WzAsMyxbMSwxLCJBIl0sWzAsMCwiXFxtYXRoc2NyIERfaSJdLFsyLDAsIlxcbWF0aHNjciBEX2oiXSxbMSwyLCJcXG1hdGhzY3IgRF97XFxhbHBoYX0iLDJdLFsxLDAsIlxccHNpX2kiLDJdLFsyLDAsIlxccHNpX2oiXV0=

Important examples of cones are the Limits and colimits of a diagram, which are called universal cones.

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