Faithful functor

This article defines a functor property: a property that can be evaluated to true/false given a functor between two categories.
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Definition

Symbol-free definition

A functor is faithful if the induced map on the collection of morphisms between any pair of objects is injective.

Definition with symbols

Suppose ${\mathcal {F}}:{\mathcal {C}}\to {\mathcal {D}}$ is a functor between two categories. We say that ${\mathcal {F}}$ is faithful if it is true that for any objects $A,B\in {\mathcal {C}}$, the induced map:

${\mathcal {F}}:{\mathcal {C}}(A,B)\to {\mathcal {D}}(FA,FB)$

is injective.