Category Theory (updating)

Definition of a category Objects and arrows exist. There should be identity arrows for all objects. Arrows should be composited. Associativity holds. The arrows are also called morphisms. Terminology: domain and codomain $\mathrm{dom}(f)\xrightarrow{f} \mathrm{cod}(f)$ Directed graphs and categories If $f:A\rightarrow B$ and $g:B\rightarrow C$ exist but $h := g\cdot f$ doesn’t, it violates the rule of composition. Hence the graphs are not categories in general. Initial object An initial object of a category $C$ is an object $I$ in $C$ such that for every object $X$ in $C$, there exists precisely one morphism $I \rightarrow X$.