Definition:Composition of Morphisms

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Let $\mathbf C$ be a metacategory.

Let $\left({g, f}\right)$ be a pair of composable morphisms.

Then the composition of $f$ and $g$ is a morphism $g \circ f$ of $\mathbf C$ subject to:

$\operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f$
$\operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g$

This composition of morphisms can be thought of as an abstraction of both composition of mappings and transitive relations.

Also see