Let \({\cal I}\) and \({\cal J}\) be small categories. Let \({\cal C}\) be a locally small category with limits of shape \({\cal I}\) and of shape
\({\cal J}\).
Define
\[\begin {array}{llll}
\mathscr {D}^{\bullet }: & {\cal I} & \to & {[{\cal J}, {\cal C}]} \\
& I & \mapsto & \mathscr {D}(I,-)
\end {array}\]
and
\[\begin {array}{rrrr}
\mathscr {D}_{\bullet }: & {\cal J} & \to & {[{\cal I}, {\cal C}]} \\
& J & \mapsto & \mathscr {D}(-, J)
\end {array}\]
Then for all \(\mathscr {D}: {\cal I} \times {\cal J} \to {\cal C}\), we have
\[
\lim _{\leftarrow {\cal J}} \lim _{\leftarrow {\cal I}} \mathscr {D}^{\bullet } \cong \lim _{\leftarrow {\cal I} {\cal J}} \mathscr {D} \cong \lim _{\leftarrow {\cal I} \leftarrow {\cal J}} \mathscr {D}_{\bullet }
\]
and all these limits exist. In particular, \({\cal C}\) has limits of shape \({\cal I} \times {\cal J}\).
