Lemma 4.19.4. Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every pair of objects $x, y$ of $\mathcal{I}$ there exist an object $z$ and morphisms $x \to z$ and $y \to z$. Then

If $M$ and $N$ are diagrams of sets over $\mathcal{I}$, then $\mathop{\mathrm{colim}}\nolimits (M_ i \times N_ i) \to \mathop{\mathrm{colim}}\nolimits M_ i \times \mathop{\mathrm{colim}}\nolimits N_ i$ is surjective,

in general colimits of diagrams of sets over $\mathcal{I}$ do not commute with finite nonempty products.

**Proof.**
Proof of (1). Let $(\overline{m}, \overline{n})$ be an element of $\mathop{\mathrm{colim}}\nolimits M_ i \times \mathop{\mathrm{colim}}\nolimits N_ i$. Then we can find $m \in M_ x$ and $n \in N_ y$ for some $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ such that $m$ maps to $\overline{m}$ and $n$ maps to $\overline{n}$. See Section 4.15. Choose $a : x \to z$ and $b : y \to z$ in $\mathcal{I}$. Then $(M(a)(m), N(b)(n))$ is an element of $(M \times N)_ z$ whose image in $\mathop{\mathrm{colim}}\nolimits (M_ i \times N_ i)$ maps to $(\overline{m}, \overline{n})$ as desired.

Proof of (2). Let $G$ be a non-trivial group and let $\mathcal{I}$ be the one-object category with endomorphism monoid $G$. Then $\mathcal{I}$ trivially satisfies the condition stated in the lemma. Now let $G$ act on itself by translation and view the $G$-set $G$ as a set-valued $\mathcal{I}$-diagram. Then

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {I} G \times \mathop{\mathrm{colim}}\nolimits _\mathcal {I} G \cong G/G \times G/G \]

is not isomorphic to

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {I} (G \times G) \cong (G \times G)/G \]

This example indicates that you cannot just drop the additional condition Lemma 4.19.2 even if you only care about finite products.
$\square$

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