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The following is Exercise 3.7 from Aluffi's Algebra: Chapter Zero (available here):

A subcategory $C'$ of a category $C$ consists of a collection of objects of $C$, with morphisms $\operatorname{Hom}_{C'} (A,B) \subseteq \operatorname{Hom}_C (A,B)$ for all objects $A, B$ in $\operatorname{Obj}(C')$, such that identities and compositions in $C$ make $C'$ into a category. A subcategory $C'$ is full if $\operatorname{Hom}_{C'} (A,B) = \operatorname{Hom}_C (A,B)$ for all $A,B$ in $\operatorname{Obj}(C')$. Construct a category of infinite sets and explain how it may be viewed as a full subcategory of set.

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  • $\begingroup$ Yes, that works well. What have you tried? There is almost nothing to do. $\endgroup$ Apr 9, 2013 at 6:41
  • $\begingroup$ That is the question. I am confused because the question does almost all the work $\endgroup$ Apr 9, 2013 at 6:42
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    $\begingroup$ On second thoufgt, you are right. One would actually define the category of infinite set as the full subcategory of Set whose objects are the infinite sets ... Hm, just write that down, i.e. "Let $Obj(C')$ be ... and for infinite sets $A,B$ let $\operatorname{Hom}_{C'}(A,B)=\operatorname{Hom}_{\mathbf {Set}}(A,B)$ ..." $\endgroup$ Apr 9, 2013 at 6:44
  • $\begingroup$ Let $Obj(C')$ be what? $\endgroup$ Apr 9, 2013 at 6:58
  • $\begingroup$ Let $Obj(C')$ be the class of infinite sets. $\endgroup$
    – Berci
    Apr 9, 2013 at 13:47

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In general, if $C$ is a category and $S \subseteq \mathrm{ob}(C)$ is a subclass of its objects, there is a unique full subcategory $C' \subseteq C$ with $\mathrm{ob}(C')=S$. This is trivial - nothing has to be proven. In the "exercise", $C$ is the category of sets, and $S$ is the class of all infinite sets. We get the category of infinite sets with maps betweem them.

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  • $\begingroup$ Done? Is that all to the exercise $\endgroup$ Apr 10, 2013 at 2:15

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