Lemma 21.2.10. Let \(I\), \(J\) and \(C\) be \(\infty \)-categories such that \(C\) admits all \(I\)-indexed and \(J\)-indexed colimits. Then \(C\) admits all \((I \times J)\)-indexed colimits, and for a functor \(F\colon I \times J \to C\) we have \[ \colim _{(i,j) \in I \times J} F(i,j) \cong \colim _{i \in I} \colim _{j \in J} F(i,j). \]
Proof. The adjunction \(\colim _J \colon \Fun (J,C) \rightleftarrows C \noloc \const \) induces an adjunction \[ \Fun (I \times J, C) \simeq \Fun (I,\Fun (J,C)) \rightleftarrows \Fun (I,C) \] by applying Lemma 21.1.6. Composing this with the adjunction \(\colim _I \colon \Fun (I,C) \rightleftarrows C \noloc \const \) then gives the result. □
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