I noticed a mathematical technique twice recently: When we want to know more about an object $X$, a strategy is to make use of Hom$(X,-)$. This methods arised in two cases.
Kodaira embedding theorem: Let $X$ be a compact Kahler manifold with a positive holomorphic line bundle. Then $X$ is holomorphically embeddable into CP$^n$. The proof simply uses the "richness" of $V := H^0(X;\mathcal{O}(F))$ (a finite dimensional vector space by the Hodge's theorem) for some line bundle $F$, taking a basis $\{s_0,...,s_N\}$ of the vector space $V$, and construct the embedding $\phi_{|F|}: X \to P^{\,N}$ by taking $x$ to $[s_0(x):...:s_N(x)]$. Still, some arguments are needed, but for me the whole point is that $H^0(X;\mathcal{O}(F))$ is "large enough".
Peter-Weyl theorem: Let $G$ be a topologically (locally) compact group. While we cannot expect there is always a faithful finite dimensional representation of $G$, there are still "enough" many finite dimensional representation: For any $x\neq y \in G$, there exists a finite dimensional representation $\phi: G\to GL(V)$ such that $\phi(x)\neq\phi(y)$. It is well-known that this theorem has lots of powerful corollaries, and is perhaps the most important basis for pur investigation on (locally) compact groups. Here the same technique appears again, namely using that Rep$(G)$ is "large enough" to make progress.
I believe such techniques must arise quite often in mathematics, and I would like to know if there are some other such examples. Thank you very much.