# 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 follow 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 follow that $$X \cong Y$$?

• mathoverflow.net/a/269745 seems to gesture towards an (unpublished) answer Sep 29, 2020 at 20:21
• after a bit of digging, I discovered that a sufficient condition based on similar factorization properties is proven in A. Pultr: Isomorphism types of objects in categories determined by numbers of morphisms. Acta Sci. Math. Szeged35(1973), 155–160 Sep 29, 2020 at 20:32
• @martti : thanks for the very useful references. In fact I had a similar idea of proof based on factorization. So it seems that it is true for instance for $C = Set_f^G$ for any finite category $G$ (as well as for their dual). Sep 30, 2020 at 18:13