It is worth mentioning that category theory can be articulated through and founded on sets, in a relatively straightforward manner.

That said, category theory and set theory seem to be two sides of the same coin even at a research level. I asked about this comparison here and got some excellent discussion from category and set theorists (see the comments).

Category theory is a universal language for discussing the notion of structure on sets, and a universal setting in which connections can be seen between seemingly disparate areas of mathematics.

One connection is mentioned in the answers above, and actually links sets to categories -- the Yoneda embedding

$${\bf Hom}_\mathcal{C}(-,\ \ ):\mathcal{C}\to{\sf Set}^{\mathcal{C}^{op}}.$$

The Yoneda Lemma shows that this embedding is fully faithful, meaning that for any category $\mathcal{C}$ there is a 'picture' of $\mathcal{C}$ inside its category of presheaves ${\sf Set}^{\mathcal{C}^{op}}$, a presheaf being a functor from its opposite category into the category of sets and functions. This presheaf category has many desirable properties it inherits from ${\sf Set}$ like (co-)completeness; it actually inherits all the structure of a topos (a very specific and nice kind of category). We see immediately that sets and categories are not separate competing entities, but different pieces of the same whole.

Other well-known examples of categorical connections between seemingly disparate areas include the duality between the category of Stone spaces and the category of Boolean algebras, or locales and frames.

The notion of a category can actually be viewed as a sufficiently general way to structure a set so that it can emulate (and generalize) many standard structured sets. Groups, rings, vector fields, modules, quotient spaces, partially ordered sets, Boolean algebras and more can all be cast as certain types of categories, and then generalized to groupoids etc. -- random variables, observables, probability measures, and states can be understood as arrows in a certain category (see here for a reference on the probability claims).

So, for someone with a background in working with structured sets, a category can be viewed as a pair of sets together with some functions between them, structured in a very general way that allows them to emulate almost all other notions of 'structure on a set'. Further, this same notion of a category can then unify these structured sets into greater structures and explore connections between them on a larger scale.

It is worth mentioning that category theory can be articulated through and founded on sets, in a relatively straightforward manner.

That said, category theory and set theory seem to be two sides of the same coin even at a research level. I asked about this comparison in this MO question and got some excellent discussion from category and set theorists (see the comments).

Category theory is a universal language for discussing the notion of structure on sets, and a universal setting in which connections can be seen between seemingly disparate areas of mathematics.

One connection is mentioned in the answers above, and actually links sets to categories -- the Yoneda embedding

$${\bf Hom}_\mathcal{C}(-,\ \ ):\mathcal{C}\to{\sf Set}^{\mathcal{C}^{op}}.$$

The Yoneda Lemma shows that this embedding is fully faithful, meaning that for any category $\mathcal{C}$ there is a 'picture' of $\mathcal{C}$ inside its category of presheaves ${\sf Set}^{\mathcal{C}^{op}}$, a presheaf being a functor from its opposite category into the category of sets and functions. This presheaf category has many desirable properties it inherits from ${\sf Set}$ like (co-)completeness; it actually inherits all the structure of a topos (a very specific and nice kind of category). We see immediately that sets and categories are not separate competing entities, but different pieces of the same whole.

Other well-known examples of categorical connections between seemingly disparate areas include the duality between the category of Stone spaces and the category of Boolean algebras, or locales and frames.

The notion of a category can actually be viewed as a sufficiently general way to structure a set so that it can emulate (and generalize) many standard structured sets. Groups, rings, vector fields, modules, quotient spaces, partially ordered sets, Boolean algebras and more can all be cast as certain types of categories, and then generalized to groupoids etc. -- random variables, observables, probability measures, and states can be understood as arrows in a certain category (see Frič, R., Papčo, M. A Categorical Approach to Probability Theory, Stud Logica 94 (2010) pp 215–230. for a reference on the probability claims).

So, for someone with a background in working with structured sets, a category can be viewed as a pair of sets together with some functions between them, structured in a very general way that allows them to emulate almost all other notions of 'structure on a set'. Further, this same notion of a category can then unify these structured sets into greater structures and explore connections between them on a larger scale.

It is worth mentioning that category theory can be articulated through and founded on sets, in a relatively straightforward manner.

That said, category theory and set theory seem to be two sides of the same coin even at a research level. I asked about this comparison here and got some excellent discussion from category and set theorists (see the comments).

Category theory is a universal language for discussing the notion of structure on sets, and a universal setting in which connections can be seen between seemingly disparate areas of mathematics.

One such connection is mentioned in the answers above, and actually links sets to categories -- the Yoneda embedding

$${\bf Hom}_\mathcal{C}(-,\ \ ):\mathcal{C}\to{\sf Set}^{\mathcal{C}^{op}}.$$

The Yoneda Lemma shows that this embedding is fully faithful, meaning that for any category $\mathcal{C}$ there is a 'picture' of $\mathcal{C}$ inside its category of presheaves ${\sf Set}^{\mathcal{C}^{op}}$, a presheaf being a functor from its opposite category into the category of sets and functions. This presheaf category has many desirable properties it inherits from ${\sf Set}$ like (co-)completeness; it actually inherits all the structure of a topos (a very specific and nice kind of category). We see immediately that sets and categories are not separate competing entities, but different pieces of the same whole.

Other well-known examples of categorical connections between seemingly disparate areas include the duality between the category of Stone spaces and the category of Boolean algebras, or locales and frames.

The notion of a category can actually be viewed as a sufficiently general way to structure a set so that it can emulate (and generalize) many standard structured sets. Groups, rings, vector fields, modules, quotient spaces, partially ordered sets, Boolean algebras and more can all be cast as certain types of categories, and then generalized to groupoids etc. -- random variables, observables, probability measures, and states can be understood as arrows in a certain category (see here for a reference on the probability claims).

So, for someone with a background in working with structured sets, a category can be viewed as a pair of sets together with some functions between them, structured in a very general way that allows them to emulate almost all other notions of 'structure on a set'. Further, this same notion of a category can then unify these structured sets into greater structures and explore connections between them on a larger scale.

It is worth mentioning that category theory can be articulated through and founded on sets, in a relatively straightforward manner.

That said, category theory and set theory seem to be two sides of the same coin even at a research level. I asked about this comparison here and got some excellent discussion from category and set theorists (see the comments).

Category theory is a universal language for discussing the notion of structure on sets, and a universal setting in which connections can be seen between seemingly disparate areas of mathematics.

One connection is mentioned in the answers above, and actually links sets to categories -- the Yoneda embedding

$${\bf Hom}_\mathcal{C}(-,\ \ ):\mathcal{C}\to{\sf Set}^{\mathcal{C}^{op}}.$$

The Yoneda Lemma shows that this embedding is fully faithful, meaning that for any category $\mathcal{C}$ there is a 'picture' of $\mathcal{C}$ inside its category of presheaves ${\sf Set}^{\mathcal{C}^{op}}$, a presheaf being a functor from its opposite category into the category of sets and functions. This presheaf category has many desirable properties it inherits from ${\sf Set}$ like (co-)completeness; it actually inherits all the structure of a topos (a very specific and nice kind of category). We see immediately that sets and categories are not separate competing entities, but different pieces of the same whole.

Other well-known examples of categorical connections between seemingly disparate areas include the duality between the category of Stone spaces and the category of Boolean algebras, or locales and frames.

The notion of a category can actually be viewed as a sufficiently general way to structure a set so that it can emulate (and generalize) many standard structured sets. Groups, rings, vector fields, modules, quotient spaces, partially ordered sets, Boolean algebras and more can all be cast as certain types of categories, and then generalized to groupoids etc. -- random variables, observables, probability measures, and states can be understood as arrows in a certain category (see here for a reference on the probability claims).

So, for someone with a background in working with structured sets, a category can be viewed as a pair of sets together with some functions between them, structured in a very general way that allows them to emulate almost all other notions of 'structure on a set'. Further, this same notion of a category can then unify these structured sets into greater structures and explore connections between them on a larger scale.