The following definition appears in Kunen (2nd edition):
For any sets $I,J$ and cardinal $\lambda$: $\text{Fn}_{\lambda}(I,J)$ is the set of all $p\in{[I\times{J}]}^{<\lambda}$ such that $p$ is the graph of a function. We make $\text{Fn}_{\lambda}(I,J)$ in to a forcing poset by letting $\leq$ be $\supseteq$ and $1=\emptyset$.
My question is: Since $p$ is a function to begin with, doesn't this mean that $p$ is also the graph of a function with $p=\{(x,p(x)):x\in{\text{dom}{(p)}}\}$? Or is there something to be gained by (possibly in a different way) regarding $p$ as the graph of a function?
I conjecture you've confused $[I\times J]^{<\lambda}$, which is the set of $<\lambda$-sized subsets of $I\times J$, with $(I\times J)^{<\lambda}$, which is a set of functions.