Definition:Injective on Morphisms

Definition

Let $\mathbf C$ and $\mathbf D$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ be a functor.

Then $F$ is said to be injective on morphisms if and only if for all morphisms $f, g$ of $\mathbf C$:

$F f = F g$ implies $f = g$

Note that it is not required that $f$ and $g$ have equal domains or codomains.

Also see

• Definition:Injective on Objects
• Definition:Surjective on Morphisms
• Definition:Embedding of Categories
• Definition:Faithful Functor

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.): Chapter $7$ Naturality: $\S 7.1$ Category of Categories: Definition $7.1$