If we want to specify the limit binary product of $A$ and $B$ in ${\bf C}$, then take ${\bf D}$ the discrete discrete category with only two objects $X_1$ and $X_2$ and no non-trivial arrows. A cone is any object $N$, which has two maps $\psi_A:N\to A$ and $\psi_B:N\to B$. But if there is a limit cone, let's call it $L\equiv A\times B$, then you can put those together to define a map $u(n):=\langle\psi_A(n),\psi_B(n)\rangle$ from $N$ to $A\times B$. And then $\psi_A(n)=\pi_1(u(n))$ and $\psi_B(n)=\pi_2(u(n))$.

While the category ${\bf C}$ might have a billion ways to look at $A$ and $B$, category theory works out that these will always just be some arrow concatenated with the limit cones binoculars - that's sort of the “why” answer to why projection operators are an ubiquitous concept.

Wikipedia: Product (category theory)

Functor

Limit . category theory