Cone

Remarks

A cone with apex $N$ is the same as a natural transformation $\Delta N \Rightarrow D$, where $\Delta N \colon J \to \mathcal{C}$ is the constant functor with value $N$; the condition $D(u) \circ \psi_j = \psi_k$ is its naturality square. Cones over $D$ and their morphisms form a category, written $\mathrm{Cone}(D)$.

Example: let $D$ be the diagram $f, g \colon X \rightrightarrows Y$ of two parallel functions between sets. Let $N = \set{x \in X : f(x) = g(x)}$, let $\psi_X$ be the inclusion and let $\psi_Y = f \circ \psi_X$. The conditions $f \circ \psi_X = \psi_Y$ and $g \circ \psi_X = \psi_Y$ hold because $f$ and $g$ agree on $N$, so this is a cone.

Non-example: in the same diagram take $N = X$, $\psi_X = \id_X$ and $\psi_Y = f$. If $f \neq g$, then $g \circ \psi_X = g \neq \psi_Y$, so the condition fails for the morphism $g$ and this is not a cone.