# When is an object determined by the number of maps from the other objects?

Let $$C$$ be a category with finite hom-sets.
Suppose that $$X$$ and $$Y$$ are objects in $$C$$ such that $$C(Z,X)cong C(Z,Y)$$ for any Z (with no naturality condition).
For which categories $$C$$ does it follows that $$X cong Y$$?
(Of course, it is true for posets).

A somewhat related question is the following.

Let $$C$$ be a symmetric monoidal closed category.
Suppose that $$X$$ and $$Y$$ are objects in $$C$$ such that $$(X,Z)cong (Y,Z)$$ for any Z (with no naturality condition).
For which categories $$C$$ does it follows that $$X cong Y$$?