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Here's a categorical proof/construction of the geometric realization of a simplicial set as left adjoint to the singular simplicial set.

I will not give a single theorem of category theory but I'll use general categorical facts to do this - this construction is very important in algebraic topology, and seeing that it actually follows from "abstract nonsense" can be illuminating. The beginning of my answer is mostly about topology; and my answer is essentially one big example.

Recall that a simplicial set is a functor $\Delta^{op}\to \mathbf{Set}$, where $\Delta$ is the category of nonempty finite ordinals and weakly order-preserving maps. Expressed this way it doesn't seem very natural and it looks like I'm using categories for categories, which may not be what you're looking for. But actually simplicial sets are combinatorial abstractions of geometric simplicial complexes, e.g. a triangle $\Delta^2$.

The basic example is the following (and the one I'm interested about here) : given a space $X$, you have a nice simplicial set associated to it, the singular simplicial set. It's given on objects by $n\mapsto \hom_{\mathbf{Top}}(\Delta^n, X)$ where $\Delta^n$ is the geometric $n$-simplex, i.e. the convex hull of the canonical basis of $\mathbb{R}^{n+1}$ (and $\hom_{\mathbf{Top}}(A,B)$ is the set of continuous maps $A\to B$). So here an $n$-simplex of this simplicial set is a geometric $n$-simplex living inside $X$. Let's denote this simplicial set by $Sing(X)$.

For nice spaces, $Sing(X)$ contains most of the information we want about $X$, homotopy-theoretically. What does this mean? It means that, in some sense we can reverse the process: start from a simplicial set $S$, create a space $|S|$, such that for spaces $X$, $|Sing(X)|$ "looks like" $X$, and "looks a lot like" $X$ for nice spaces $X$. This is what we want geometric realization to mean.

So we want $Sing(X)$ to be the simplicial set "best approximating $X$" (or $|S|$ to be the space best approximating $S$). It turns out that that's what adjoint functors do very often.

To start with the categorical stuff, let's see what we can try to do. We want $|-|$ to be adjoint to $Sing$: but which adjoint, left or right ? Well it's very easy to see that $Sing$ preserves limits but not colimits in general, so if it is to be adjoint, it must be a right adjoint, hence $|-|$ must be its left adjoint (here we use the general fact that adjoints preserve (co)limits, depending on which adjoint they are).

Given a simplicial set $Y$, let's compute $\hom(Y,Sing(X))$. A morphism of simplicial sets is a natural transformation by definition, so this is $Nat(Y,Sing(X))$.

By general categorical stuff, this is isomorphic (naturally in $Y$ and $X$) to $\displaystyle\int_{n\in \Delta^{op}} \hom(Y_n, Sing(X)_n)$ (the end). But $\hom(Y_n, Sing(X)_n) = \hom (Y_n, \hom_{\mathbf{Top}}(\Delta^n, X))= \displaystyle\prod_{y\in Y_n}\hom_{\mathbf{Top}}(\Delta^n, X)$, which is, because $\mathbf{Top}$ has coproducts, naturally isomorphic to $\hom_{\mathbf{Top}}(\displaystyle\coprod_{y\in Y_n}\Delta^n, X)$. Now an easy computation in $\mathbf{Top}$ shows that $\displaystyle\coprod_{y\in Y_n}\Delta^n\cong Y_n\times \Delta^n$ where $Y_n$ has the discrete topology; therefore $\hom(Y, Sing(X)) \cong \displaystyle\int_{n\in\Delta^{op}} \hom_{\mathbf{Top}}(Y_n\times \Delta^n, X)$ .

A general categorical theorem about ends and coends tells us that, since $\mathbf{Top}$ is cocomplete, it has all coends, and $\displaystyle\int_{n\in\Delta^{op}} \hom_{\mathbf{Top}}(Y_n\times \Delta^n, X)\cong \hom_{\mathbf{Top}}(\displaystyle\int^{n\in \Delta^{op}} Y_n\times \Delta^n, X)$.

This tells us that $Y\mapsto \displaystyle\int^{n\in \Delta^{op}} Y_n\times \Delta^n$ is a left adjoint to $Sing$. How do we know we haven't made a specific choice ? Well the Yoneda lemma tells us that adjoints are unique up to isomorphism, so if you have another choice for $|-|$, it's actually essentially the same as mine.

But the point is that we know how to compute coends in $\mathbf{Top}$, so we now have an explicit description of $|Y|$: essentially we take one $n$-simplex per element $y\in Y_n$ and we glue these simplices together according to the degeneracy and face maps of $Y$.

So we used general knowledge about ends and coends, coproducts, and the Yoneda Lemma to prove that we have a geometric realization that satisfies our requirements, and that it's the only one that can do so.