A square with monic legs is a pullback exactly when it identifies the source with the intersection subobject

By [L1], the pullback $P\to D$ of the two monics $m$ and $n$ exists. Because the square defining $P$ is a common lower bound of $[m]$ and $[n]$, and every other common lower bound factors uniquely through that pullback, [L2] says that $P\to D$ represents their intersection subobject.

If the displayed square is a pullback, then its source $X\to D$ is another representative of the same greatest lower bound from step 1.1. Therefore the induced morphism $X\to I$ is an isomorphism.

Conversely, if the induced map $X\to I$ is an isomorphism, then composing the pullback square representing $I$ from step 1.1 with that isomorphism yields the displayed square. Pullbackness is invariant under replacing the corner object by an isomorphic one, so the displayed square is a pullback.