Definition (cone):

Let ${\displaystyle {\mathcal {C}}}$ be a category and let ${\displaystyle G=(V,A)}$ be a diagram in ${\displaystyle {\mathcal {C}}}$. A cone over ${\displaystyle G}$ is an object ${\displaystyle a}$ of ${\displaystyle {\mathcal {C}}}$, together with morphisms ${\displaystyle \phi _{a,b}:a\to b}$ for each ${\displaystyle b\in V}$, so that for each ${\displaystyle \psi :b\to b'\in A}$ (so that ${\displaystyle b,b'\in V}$) we have

${\displaystyle \phi _{a,b'}=\psi \circ \phi _{a,b}}$.

Definition (limit):

Let ${\displaystyle {\mathcal {C}}}$ be a category, let ${\displaystyle G=(V,A)}$ be a diagram in ${\displaystyle {\mathcal {C}}}$ and let

Definition (diagonal):

Let ${\displaystyle {\mathcal {C}}}$ be a category, and let ${\displaystyle X}$ be an object of ${\displaystyle {\mathcal {C}}}$. Further, suppose that the object ${\displaystyle X\times X}$ and the morphisms ${\displaystyle \pi _{1}:X\times X\to X}$ and ${\displaystyle \pi _{2}:X\times X\to X}$ constitute a product of ${\displaystyle X}$ with itself in ${\displaystyle {\mathcal {C}}}$. Then the diagonal (also called "diagonal morphism") is the unique morphism

${\displaystyle \Delta :X\to X\times X}$

such that ${\displaystyle \pi _{1}\circ \Delta =\pi _{2}\circ \Delta =\operatorname {Id} _{X}}$.

Definition (anti-diagonal):

Let ${\displaystyle {\mathcal {C}}}$ be a category, and let ${\displaystyle X}$ be an object of ${\displaystyle {\mathcal {C}}}$. Further, suppose that the object ${\displaystyle X\sqcup X}$ and the morphisms ${\displaystyle \iota _{1}:X\to X\sqcup X}$ and ${\displaystyle \iota _{2}:X\to X\sqcup X}$ constitute a coproduct of ${\displaystyle X}$ with itself in ${\displaystyle {\mathcal {C}}}$. Then the anti-diagonal (also called "anti-diagonal morphism") is the unique morphism

${\displaystyle \nabla :X\sqcup X\to X}$

such that ${\displaystyle \nabla \circ \iota _{1}=\nabla \circ \iota _{2}=\operatorname {Id} _{X}}$.