I was trying to figure out what is the universal property of Grassmannians, I saw a similar question Universal property of the Grassmanian [closed].
And I have checked the references in the answer, and those references including Wiki didn't state the universal property formally. For example, Eisenbud & Harris' book shows the functorial of the Grassmannian $Gr_S(k,n)$ over the base scheme $S$. And three ways to construct the Grassmannian: locally covered by affine schemes, using Plucker equations embedded into projective space, as Hilbert scheme.
What does the universal property of Grassmannian really mean?
In [Gortz and Wedhorn section 8.4], the Grassmanian is a representable functor from $\text{Sch}^{\text{opp}} \to \text{Sets}$, such that
$$\text{Grass}_{d,n}(S) = \{\mathcal{U} \subset \mathcal{O}^n_S \mid \mathcal{O}^n_S/\mathcal{U}\text{ is locally free of rank } n-d\}$$
Which is represented by the smooth scheme Grassmannian denoted it also $X = \text{Grass}_{d,n} = G(d,n)$.
Now we have $$h_X(-) \cong \text{Grass}_{d,n}(-)$$ with $id\in h_X(X)$ sends to the tautological bundle $\gamma \in \text{Grass}_{d,n}(X)$ over $X$ via the isomorphism above.
Given any scheme $S$ with some subbundle $\mathcal{E}$ of trivial bundle then we have $\mathcal{E} \in \text{Grass}_{d,n}(S)$, therefore exist some $f\in h_X(S) = \text{Hom}(S,G(d,n))$ such that $f^*\gamma = \mathcal{E}$ which coinside with what shown in the comment