Is an equivalence between categories injective on objects

Question

Roughly speaking, a functor between two categories is said to be an equivalence if it is both fully faithful and is surjective on objects up to isomorphism. Does it follow from this definition that an equivalence is injective on objects? Is each equivalence an embedding?

Answer 1

Let's take a trivial example: $C_1$ a category with one object and no morphisms except the identity, and $C_2$ a category with two objects and with each hom-set having one element. There's a unique functor $F:C_2\to C_1$. It is an equivalence of categories.