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Is the composition in locally small categories always the set-theoretic composition of functions?

The definition of a locally small category simply says:

A category $\mathcal{C}$ is called locally small if for any $X, Y \in \text{ob}\mathcal{C}$ we have that $\text{Hom}_\mathcal{C}(X,Y)$ is a set.

But is it always implicitly assumed that for $f,g \in \text{Mor}\mathcal{C}$ we have $fg = f \circ_{set} g$?

I also think that this assumption is used to prove that the $\text{Hom}$ functor of a locally small category is actually a functor.

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  • $\begingroup$ No: there is no assumption that morphisms are functions. $\endgroup$ Commented Apr 9, 2019 at 10:10

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Locally small just says that the $Hom_\mathcal{C}(X, Y)$ is a set, it tells us nothing about what might be in that set. For many categories we naturally consider, this is actually the set of maps between certain mathematical structures (e.g. group homomorphisms, continuous maps, etc). This does not always need to be the case.

For example, any poset $P$ can be considered as a category and it is definitely locally small. However, the composition of $x \leq y$ and $y \leq z$ is just $x \leq z$.

Another example would be any group $G$. We can consider $G$ as a category with one object $X$ and a morphism $X \to X$ for every element in the group. Composition is then given by the group operation.

More generally, there is a notion close to what you are describing. That of a concrete category (Wikipedia, nLab). This is a category $\mathcal{C}$ equipped with a faithful functor $U: \mathcal{C} \to \mathbf{Set}$. A very natural example is the forgetful functor on for example the category of groups, or the category of topological spaces.

This idea also gives another non-example to your question: the category of topological spaces where the morphisms are homotopy classes of continuous functions. It turns out that this category does not admit a faithful functor into $\mathbf{Set}$ and can thus not be viewed as a concrete category.

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