# 10.1.1 Limits and colimits

Since sets are closed under^{} products^{}, the universal property^{} of products in \autorefthm:prod-ump shows immediately that $\mathcal{S}et$ has finite products.
In fact, infinite products follow just as easily from the equivalence

$$(X\to \prod _{a:A}B(a))\simeq \left(\prod _{a:A}(X\to B(a))\right).$$ |

And we saw in \autorefex:pullback^{} that the pullback of $f:A\to C$ and $g:B\to C$ can be defined as ${\sum}_{(a:A)}{\sum}_{(b:B)}f(a)=g(b)$; this is a set if $A,B,C$ are and inherits the correct universal property.
Thus, $\mathcal{S}et$ is a *complete ^{}* category

^{}in the obvious sense.

Since sets are closed under $+$ and contain $\mathrm{\U0001d7ce}$, $\mathcal{S}et$ has finite coproducts^{}.
Similarly, since ${\sum}_{(a:A)}B(a)$ is a set whenever $A$ and each $B(a)$ are, it yields a coproduct of the family $B$ in $\mathcal{S}et$.
Finally, we showed in \autorefsec:pushouts that pushouts exist in $n$-types, which includes $\mathcal{S}et$ in particular.
Thus, $\mathcal{S}et$ is also *cocomplete*.

Title | 10.1.1 Limits and colimits^{} |
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