Preserves and Reflects

Given two categories \( \mathcal{C},\mathcal{D} \), and a functor \( F : \mathcal{C} \rightarrow \mathcal{D} \), we say that \( F \)

• preserves a propery \( P \) if whenever an object or arrow, \( c \), of \( \mathcal{C} \) has \( P \), then so does \( F(c) \).
• reflects a property \( P \) if whenever an object or arrow, \( d = F(c) \), of \( \mathcal{D} \) has \( P \), then so does \( c \).

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