[[Category theory MOC]]
# Zero morphism

In a category $\cat C$, a **constant morphism** $c \in \cat C(X,Y)$ satisfies $cf = cg$ for any $f,g \in \cat C(Z,X)$ and $Z \in \cat C$,
whereas a **coconstant morphism** $c \in \cat C(X,Y)$ satisfies $fc = gc$ for any $f,g \in \cat C(Y,Z)$.
A **zero morphism** is both a constant and coconstant morphism. #m/def/cat 

A category $\cat C$ is said to **have zero morphisms** iff for any two objects $X,Y \in \cat C$ there is a fixed morphism $0_{XY} \in \cat C(X,Y)$ such that the following diagram commutes #m/def/cat 

<p align="center"><img align="center" src="https://i.upmath.me/svg/%0A%5Cusetikzlibrary%7Bcalc%7D%0A%5Cusetikzlibrary%7Bdecorations.pathmorphing%7D%0A%5Ctikzset%7Bcurve%2F.style%3D%7Bsettings%3D%7B%231%7D%2Cto%20path%3D%7B(%5Ctikztostart)%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%3DWzAsNCxbMCwwLCJYIl0sWzAsMiwiWSJdLFsyLDAsIlgiXSxbMiwyLCJZIl0sWzAsMSwiZiIsMl0sWzIsMywiZyJdLFswLDIsIjBfe1hYfSJdLFsxLDMsIjBfe1lZfSIsMl0sWzAsMywiMF97WFl9IiwxXV0%3D%0A%5Cbegin%7Btikzcd%7D%5Bampersand%20replacement%3D%5C%26%5D%0A%09X%20%5C%26%5C%26%20X%20%5C%5C%0A%09%5C%5C%0A%09Y%20%5C%26%5C%26%20Y%0A%09%5Carrow%5B%22%7B0_%7BXX%7D%7D%22%2C%20from%3D1-1%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22f%22'%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%7B0_%7BXY%7D%7D%22%7Bdescription%7D%2C%20from%3D1-1%2C%20to%3D3-3%5D%0A%09%5Carrow%5B%22g%22%2C%20from%3D1-3%2C%20to%3D3-3%5D%0A%09%5Carrow%5B%22%7B0_%7BYY%7D%7D%22'%2C%20from%3D3-1%2C%20to%3D3-3%5D%0A%5Cend%7Btikzcd%7D%0A#invert" alt="https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsMiwiWSJdLFsyLDAsIlgiXSxbMiwyLCJZIl0sWzAsMSwiZiIsMl0sWzIsMywiZyJdLFswLDIsIjBfe1hYfSJdLFsxLDMsIjBfe1lZfSIsMl0sWzAsMywiMF97WFl9IiwxXV0=" /></p>

for any $X,Y \in \cat C$ and $f,g \in \cat C(X,Y)$.

## Properties

- A category with zero morphisms allows one to define the [[Kernels and cokernels]] of morphisms.


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