Easy to understand examples of category theoretic theorems that are useful

Every now and then I hear a category theorist saying that category theory is a unifying language for mathematics, and that category theory proves general theorems that some people would prove separately for each concrete case.

Can you guys give me easy to understand examples of such general theorems that some people would prove separately for each concrete case? This is because I want to get a feeling for the usefulness and powerfulness of category theory via examples. Note that general discussion on why category theory is useful was already given here.

Here's one of the first examples I personally found useful, and which I employed constantly throughout my mathematical education. The basic fact is that left adjoint functors preserve colimits and right adjoint functors preserve limits. Random examples off the top of my head:

• An extremely concrete example: the inclusion of the poset $\mathbb{Z}$ into the poset $\mathbb{R}$ has both a left and a right adjoint, given by the ceiling and floor functions respectively (exercise). This means that it preserves both limits and colimits, which for posets are meets / infima and joins / suprema respectively.
• The free group functor is a left adjoint, so sends disjoint union (the coproduct of sets) to free product (the coproduct of groups). For example, the free group $F_2$ on two generators is the free product of two copies of the free group $\mathbb{Z}$ on one generator.
• More generally, the forgetful functor from a category of algebraic objects (groups, rings, modules) to sets typically has a left adjoint, the corresponding free functor. This means that the forgetful functor preserves limits, so limits of algebraic objects, when they exist, can be computed as limits of sets.
• The inclusion of sheaves into presheaves is a right adjoint, so preserves limits. This implies that limits of sheaves, when they exist, can be computed as limits of presheaves, which are easy to compute because they're computed pointwise. (Colimits are trickier.)
• Tensor product, in most of its incarnations, is a left adjoint, so preserves colimits. For example, extension of scalars preserves direct sums and cokernels. This means it is left exact and consequently has right derived functors.

The concept of adjoint functors is useful for much more than this, though. It trains you to look for which mathematical constructions can be expressed as functors, so that you can then ask the question of whether they have adjoints, which if they exist may be interesting new mathematical constructions. In other words, the concept of adjoint functors is a fruitful source of questions in addition to answers.

Not a theorem, but a theory! The theory of Galois Categories explains why the study of covering spaces in algebraic topology and the study of algebraic field extensions is so similar. In fact, going a step further, one can also try to understand the Galois theory of differential field extensions.