Here's an example I learned from Todd Trimble. Recall that the degree $k$ part of the exterior algebra on a vector space $V$ of dimension $n$ has dimension ${n \choose k}$, and similarly the degree $k$ part of the symmetric algebra has dimension ${n+k-1 \choose k} = \left( {n \choose k} \right)$. So one can think of these constructions as categorifying binomial coefficients. More precisely, the exterior algebra categorifies its Hilbert series $(1 + t)^n$, and the symmetric algebra categorifies its Hilbert series $\frac{1}{(1 - t)^n}$.

But there's more! The duality between the Hilbert series above manifests itself in the identity $\left( {-n \choose k} \right) = (-1)^k {n \choose k}$, which categorifies to the following statement: "the exterior algebra is the symmetric algebra of a purely odd supervector space." So isomorphisms in the category of supervector spaces categorify identities involving negative binomial coefficients.