Constitution is the relation between an object and the matter that composes it, which is distinct from identity. A statue and the lump of clay that constitutes it share all the same parts at a time, yet they have different identity conditions: the statue ceases to exist when reshaped, but the lump continues. This relation must be carefully distinguished from identity to handle these cases coherently.
From your study of the identity of indiscernibles, you know Leibniz's Law: if two things are identical, they share all the same properties. Now consider a sculptor who shapes a lump of clay into a statue of David. When the statue is complete, there appears to be one spatially coincident object — the lump and the statue occupy the exact same location and are made of exactly the same matter. Are they identical? The constitution relation is the metaphysician's answer to why the answer is no.
The argument that the statue and the lump are not identical runs through their modal and temporal properties — the kind of properties you studied in substance and property theory. The statue would be destroyed if the clay were squashed into a ball; the lump would survive. The lump existed before the statue was sculpted; the statue did not. If identity is symmetric and transitive, two things with different properties cannot be identical. The statue has the property "would cease to exist if reshaped"; the lump lacks this property. By Leibniz's Law, they are not identical. Yet they share all their intrinsic, non-modal, present-tense physical properties. This is the puzzle.
Constitution is the relation that holds between the lump and the statue: the lump constitutes the statue without being identical to it. Constitution is like identity in that it requires complete material overlap at a time — the constituting object and the constituted object are co-located and share all their parts. But unlike identity, constitution is not symmetric: the lump constitutes the statue, but the statue does not constitute the lump. And it is not permanent: if the sculptor melts down the statue, the lump no longer constitutes anything, but the lump persists.
The deeper lesson is that identity conditions are object-sortal-relative. What makes something the same statue over time is different from what makes something the same lump of clay over time. Statues are typed under the sortal "statue," which brings with it specific persistence and individuation conditions — statues persist through minor repairs but not through fundamental reshaping. Lumps are typed under "lump of clay," which has different persistence conditions. Because objects can be co-located while falling under different sortals with different identity conditions, we can have two numerically distinct objects sharing a single region of space. The constitution relation is precisely what holds between objects in this situation: full material overlap, different identity conditions, different sortal kinds.
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