Definition 4.5 (Limit preserving / reflecting / creating). Let $F:C\to D$ be a functor.

• (a)
  We say $F$ preserves limits of shape $J$ if, given $D:J\to C$ and a limit cone $(L,({\lambda }_j|j\in \mathrm{ob}J))$ for it, $(FL,(F{\lambda }_j|j\in \mathrm{ob}J))$ is a limit for $FD:J\to D$.
• (b)
  We say $F$ reflects limits of shape $J$ if given $D:J\to C$, any cone over $D$ which maps to a limit cone in $D$ is a limit in $C$.
• (c)
  We say $F$ creates limits of shape $J$ if, given $D:J\to C$ and a limit cone $(L,({\lambda }_j|j\in \mathrm{ob}J))$ over $FD$, there exists a cone over $D$ whose image under $F$ is $\cong (L,({\lambda }_j))$, and any such cone is a limit in $C$.