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As we know, the Projective space P^n represent the functor sending X to the set of line bundles L on X together with a surjection from the trivial vector bundle to L.

My question is, what functor does the Grassmannian Gr(d,n) represent? Sending X to the set of rank-$n$ vector bundles together with a subbundle of rank $d$?

More generally, what functor do flag varieties represent?

As we know, the projective space $\mathbb{P}^n$ represents the functor sending $X$ to the set of line bundles $L$ on $X$ together with a surjection from the trivial vector bundle to $L$.

My question is, what functor does the Grassmannian $Gr(d,n)$ represent? Sending $X$ to the set of rank-$n$ vector bundles together with a sub-bundle of rank $d$?

More generally, what functors do flag varieties represent?

As we know, the Projective space P^n represent the functor sending X to the set of line bundles L on X together with a surjection from the trivial vector bundle to L.

My question is, what functor does the grassmannian Gr(d,n) represent? Sending X to the set of rank n vector bundles together with a subbundle of rank d?

More generally, what functor does flag varieties represent?

As we know, the Projective space P^n represent the functor sending X to the set of line bundles L on X together with a surjection from the trivial vector bundle to L.

My question is, what functor does the Grassmannian Gr(d,n) represent? Sending X to the set of rank-$n$ vector bundles together with a subbundle of rank $d$?

More generally, what functor do flag varieties represent?