Morphism in a category ($f:A\to B$)

A categorical morphism is an arrow with a specified source and target in a category. An arrow $f:A\to B$ composes with $g:B\to C$ to give $gf:A\to C$. Composition is associative, and every object has an identity arrow acting as a unit. A functor preserves the sources, targets, identities and compositions of these arrows. In a locally small category, the arrows between fixed objects form a hom-set.