- Let $\mathbf{I}$ be a small category. A functor $F{\colon}\linebreak[0] \mathscr{A} \to \mathscr{B}$ preserves limits of shape $\mathbf{I$} if for all diagrams $D{\colon}\linebreak[0] \mathbf{I} \to \mathscr{A}$ and all cones $\Bigl(A \stackrel{p_I}{\longrightarrow} D(I)\Bigr){I \in \mathbf{I}}$ on $D$, \begin{align*} & \Bigl(A \stackrel{p_I}{\longrightarrow} D(I)\Bigr){I \in \mathbf{I}} \text{ is a limit cone on } D \text{ in }\mathscr{A}\ \implies & \Bigl(F(A) \stackrel{Fp_I}{\longrightarrow} FD(I)\Bigr)_{I \in \mathbf{I}} \text{ is a limit cone on } F \circ D \text{ in }\mathscr{B}. \end{align*}\n2. A functor $F{\colon}\linebreak[0] \mathscr{A} \to \mathscr{B}$ preserves limits if it preserves limits of shape $\mathbf{I}$ for all small categories $\mathbf{I}$.\n3. Reflection of limits is defined as in~, but with $\Longleftarrow$ in place of $\Longrightarrow$.