Constant diagrams, cones, cocones, and their morphisms

Let $\mathcal{J}$ and $\mathcal{C}$ be categories and let $c\in \mathcal{C}$. The constant diagram at $c$ is the functor $\mathrm{\Delta }c:\mathcal{J}\to \mathcal{C}$ that sends every object to $c$ and every morphism to $1_c$ (Covariant functor, identity functor, composite functor, and contravariant functor).

For a diagram $D:\mathcal{J}\to \mathcal{C}$ (Diagram as a functor from an indexing category), a cone over $D$ with apex $c$ is a natural transformation $\lambda :\mathrm{\Delta }c\Rightarrow D$ (Natural transformation and its components). Thus it is a family ${\lambda }_j:c\to D(j)$ satisfying

$D(u){\lambda }_j={\lambda }_k(u:j\to k)\mathrm{.}$

A morphism of cones $(c,\lambda )\to (c^{\mathrm{\prime }},{\lambda }^{\mathrm{\prime }})$ is a morphism $h:c\to c^{\mathrm{\prime }}$ such that ${\lambda }_j^{\mathrm{\prime }}h={\lambda }_j$ for every $j$. Cones and their morphisms form the category $\mathrm{Cone}(D)$.

Dually, a cocone under $D$ with apex $c$ is a natural transformation $\rho :D\Rightarrow \mathrm{\Delta }c$, so ${\rho }_{k}D(u)={\rho }_j$. A morphism of cocones $(c,\rho )\to (c^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }})$ is a morphism $h:c\to c^{\mathrm{\prime }}$ satisfying $h{\rho }_j={\rho }_j^{\mathrm{\prime }}$ for every $j$. These form $\mathrm{Cocone}(D)$.