Morphism

Let $\mathcal C$ be a category and $X,Y \in \mathrm{Ob}(\mathcal C)$. A morphism $f : X \to Y$ is an element of $\mathrm{Hom}_{\mathcal C}(X,Y)$.

• The source/domain of $f$ is $X$ (compare domain).
• The target/codomain of $f$ is $Y$ (compare codomain).
• Morphisms can be composed when targets and sources match, and each object has an identity morphism.

Special cases:

• If $X=Y$, then $f$ is an endomorphism.
• If $f$ is invertible (has a two-sided inverse), then $f$ is an isomorphism.
• If $f$ is left-cancellative under composition, then $f$ is a monomorphism (a “mono”).
• Dually, one has epimorphisms (“epis”).