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I have to show that the subcategory of infinite sets, say $\infty$-Set, is a full subcategory. However, is this really as simple as:

Consider two objects, A and B, in $\infty$-Set. Note that $Hom_{\infty-Set}(A,B) = B^A$. Hence, it equals $Hom(A,B)$ trivially.

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    $\begingroup$ What is "the subcategory of infinite sets"? To define a subcategory you have to state which objects and which morphisms are in it. What do you mean by "full subcategory"? There are at least two definitions, one of which is not in common use but would require you to check a further condition. It seems to me that the exercise is just to check that you know the definitions. $\endgroup$
    – Zhen Lin
    May 31 at 6:08
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    $\begingroup$ @ZhenLin could you perhaps clarify what you mean? $\endgroup$
    – magma
    May 31 at 12:49

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If you consider the class of infinite sets with all functions between them, yes it is a category and indeed is a full subcategory of SET, for the reason you mentioned. Seems like a very basic exercise to me....unless I am missing something, but I doubt.

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