lemma. limits commute with limits [leinster2016basic, 6.2.8] [tt-0047]

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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 \({\cal I} \times {\cal J}\).