How can we see that the categorical limit of the diagram $F$ from the discrete 2 object category to sets is isomorphic to the usual product $\{(x,y)\mid x\in X, y\in Y\}$ defined elementwise on sets. Certainly if im$(F)=\{X,Y\}$ then $X\times Y$ is the apex of a cone over $F$ using the projection map.
I think the correct approach should be to consider sets strictly larger, smaller or equal in size to $X\times Y$ because that would allow us to construct (or perhaps fail to construct) maps through $X\times Y$ but I'm not sure how to formalise this.
Fix a set $\{*\}$ with precisely one element and recall that elements of a set $X$ are precisely the functions $\{*\}\rightarrow X$.
By definition the product set $X\times Y=\{(x,y)\mid x\in X, y\in Y\}$ satisfies that for any pair of functions $x:\{*\}\rightarrow X$, $y:\{*\}\rightarrow Y$ there exists a unique function $(x,y):\{*\}\rightarrow X\times Y$ satisfying $pr_X \circ (x,y) = x$ and $pr_Y \circ (x,y) = y$. This is to say: the set theoretic product satisfies the universal property of a categorical products for cones with apex $\{*\}$.
What about general cones with an arbitrary set $T$ as apex? Consider two functions $f:T\rightarrow X$ and $g:T\rightarrow Y$. We need to show that there exists a unique function $(f,g): T \rightarrow X\times Y$ with $pr_X\circ (f,g) = f$ and $pr_Y\circ (f,g)= g$. Now note that a function may be defined by specifying what it does on the elements of its domain. So if we know what we have to do for each element $t\in T$, we are done. Recall again that an element $t\in T$ is nothing but a function $t:\{*\}\rightarrow T$. Postcomposing with the functions $f,g$ we thus get two functions $f\circ t: \{*\}\rightarrow X$ and $g\circ t: \{*\} \rightarrow Y$, which amount by definition to the elements $f(t)\in X$ and $g(t)\in Y$. By the discussion in the first paragraph they induce a unique element $(f(t),g(t))\in X\times Y$. But this means that if we want to have a function $(f,g):T \rightarrow X\times Y$ (with given constraints) it must send $t$ to $(f(t),g(t))$. This already shows uniqueness, it is left to show that the assignment $t\mapsto (f(t),g(t))$ is indeed a welldefined function.
To recap: The set theoretic product satisfies the universal property of a categorical product, since it does for cones with apex $\{*\}$ and for general cones with apex $T$ the functions are completely determined on what the do for elements of $T$, which are functions $\{*\}\rightarrow T$.